f^-^.^1
UNITED STATES DEPARTMENT OF COMMERCE • John T. Connor, Secretary
NATIONAL BUREAU OK STANDARDS • A. V. Astin, Director
Radio Meteorology
B. R. Bean and E. J. Dutton
Central Radio Propagation Laboratory*
National Bureau of Standards
Boulder, Colorado
*The CRPL was transferred to the
Enviromental Science Services Administration
while this Monograph was in press.
National Bureau of Standards Monograph 92
Issued March 1, 1966
Yor sale by the Superintendent of Documents, U. S. Government Printing Office
Washington, D. C. 20402 Price ?2.75
Library of Congress Catalog Card No. 65-60033
Foreword
This volume brings together the work done in radio meteorology
over the past ten years at the National Bureau of Standards' Central
Radio Propagation Laboratory (CRPL). This decade has seen the
development, on an international scale, of great emphasis upon the effects
of the lower atmosphere on the propagation of radio waves. The CRPL
group has concentrated upon the refraction of radio waves as well as the
refractive index structure of the lower atmosphere on both synoptic and
climatic scales. These are the areas of radio meteorology that are treated
in this volume, with additional chapters on radio-meteorological param-
eters and the absorption of radio waves by the various constituents of the
lower atmosphere. An effort has been made to include results obtained
in other laboratories both in the United States and abroad.
A. V. AsTiN, Director.
Preface
The authors express their gratitude and appreciation to the many
members of the Central Radio Propagation Laboratory staff and others
with whom they have had the pleasure of working. This volume would
not have been possible without their assistance, encouragement, and
stimulation. Particular acknowledgements are due : R. L. Abbott, Mrs.
L. Bradley, Miss B. A. Gaboon, C. B. Emmanuel, L. Fehlhaber,* V. R.
Frank, Miss S. Gerkin, Mrs. B. J. Gibson, J. Grosskopf,* J. W. Herbstreit,
J. D. Horn, J. A. Lane,t J. R. Lebsack, R. E. McGavin, Miss F. M.
Meaney, C. M. Miller, K. A. Norton, A. M. Ozanich, Jr., the late Mrs.
G .E. Richmond, L. P. Riggs, P. L. Rice, C. A. Samson, W. B. Sweezy,
G. D. Thayer, Mrs. B. J. Weddle, E. R. Westwater, Mrs. P. C. Whittaker
and W. A. Williams.
B. R. Bean
E. J. Button
*Fernmeldetechnisches, Darmstadt, Germany.
tRadio Research Station, Slough, England.
Contents
Foreword iii
Preface iv
Chapter 1. The radio refractive index of air 1
1.1. Introduction 1
1.2. Dielectric constant of moist air 2
1.3. Constants in the equation for A'^ 4
1.4. Errors in the practical use of the equation for A^ 9
1.5. Presentation of A^ data 13
1.6. Conclusions 20
1.7. References 21
Chapter 2. Measuring the radio refractive index 23
2.1. The measurement of the radio refractive index 23
2.2. Indirect measurement of the radio refractive index 23
2.3. Direct measurement of the refractive index 30
2.4. Comparison between the direct and indirect methods of measure-
ment 38
2.5. Radiosonde lag constants 38
2.5.1. Introduction 38
2.5.2. Theory of sensor time lags 29
2.5.3. Radiosonde profile analysis 41
2.5.4. Conclusion 43
2.6. References 45
Chapter 3. Tropospheric refraction 49
3.1. Introduction 49
3.2. Limitations to radio ray tracing 52
3.3. An approximation for high initial elevation angles 53
3.4. The statistical method 54
3.5. Schulkin's method 54
3.6. Linear or effective earth's radius model 56
3.7. Modified effective earth's radius model 59
3.8. The exponential model 65
3.9. The initial gradient correction method 77
3.10. The departures-from-normal method 77
3.11. A graphical method 80
3.12. Derivations 1 82
3.13. References 87
Chapter 4, N climatology 89
4.1. Introduction 89
4.2. Radio-refractive-index climate near the ground 89
4.2.1. Introduction 89
4.2.2. Presentation of basic data 89
4.2.3. Worldwide values of A^o 98
4.2.4. Climatic classification by A^, 102
4.2.5. Applications 105
4.2.6. Critical appraisal of results 107
4.2.7. Conclusions 108
4.3. On the average atmospheric radio refractive index structure over
North America 109
4.3.1. Introduction 109
4.3.2. Meteorological data and reduction techniques 109
4.3.3. Average A'^o structure 111
4.3.4. Continental cross sections 122
4.3.5. Delineation of climatic characteristics 129
4.4. The climatology of ground-based radio ducts 132
4.4.1. Introduction 132
4.4.2. Meteorological conditions associated with radio
refractive index profiles 132
4.4.3. Refractive conditions due to local meteorological
phenomena 134
4.4.4. Background 135
4.4.5. Description of observed ground-based atmospheric
ducts 140
4.5. A study of fading regions within the horizon caused by a surface
duct below a transmitter 146
4.5.1. Introduction 146
4.5.2. Regions and extent of fading within the horizon in
the presence of superref raction 147
4.5.3. Theory and results 149
4.5.4. Sample computations 157
4.6. Air mass refractive properties 163
4.6.1. Introduction 163
4.6.2. Refraction of radio rays 164
4.6.3. Conclusions 170
4.7. References 170
Chapter 5. Synoptic radio meteorology 173
5.1. Introduction 173
5.2. Background 173
5.3. Refractive index parameters 185
5.4. A synoptic illustration 195
5.5. Surface analysis in terms of A''o 197
5.6. Constant pressure chart analysis 205
5.7. Vertical distribution of the refractive index using A units 211
5.8. Summary 223
5.9. References 224
Chapter 6. Transhorizon radio-meteorological parameters 229
6.1. Existing radio-meteorological parameters 229
6.1.1. Introduction 229
6.1.2. Parameters derived from the A'^ profile 233
6.1.3. Parameters involving thermal stability 238
6.1.4. Composite parameter 238
6.1.5. Potential refractive index (or modulus), K 239
6.1.6. Vertical motion of the atmosphere 240
6.1.7. Discussion of the parameters 241
6.1.8. Comparison of some parameters 243
6.1.9. Some exceptions and anomalies 243
6.1.10. Conclusions 246
6.2. An analysis of VHF field strength variations and refractive index
profiles 247
6.2.1. Introduction 247
6.2.2. Radio and meteorological data 247
6.2.3. Classification of radio field strengths by profile types. 248
6.2.4. Prediction of field strength for unstratified conditions. 250
6.2.5. The effect of elevated layers on the Illinois paths 251
6.2.6. Elevated layers at temperature inversions 252
6.2.7. The influence of small layers 255
6.2.7. Conclusions 257
6.3. A new turbulence parameter 258
6.3.1. Introduction 258
6.3.2. The concept of thermal stability 258
6.3.3. The adiabatic lapses of TV 261
6.3.4. The turbulence parameter, II 262
6.3.5. Comparison of radio-meteorological parameters 263
6.3.6. Conclusion 266
6.4. References 266
Chapter 7. Attenuation of radio waves 269
7.1. Introduction 269
7.2. Background 269
7.3. Attenuation by atmospheric gases 270
7.4. Estimates of the range of total gaseous absorption 280
7.5. Total radio path absorption 283
7.6. Derivation of absorption estimate for other areas 286
7.7. Attenuation in clouds 291
7.8. Attenuation by rain 292
7.9. Rainfall attenuation climatology 297
7.10. Rain attenuation effects on radio systems engineering 298
7.11. Attenuation by hail 302
7.12. Attenuation by fog 303
7.13. Thermal noise emitted by the atmosphere 304
7.14. References 308
Chapter 8. Applications of tropospheric refraction and refractive index
models 311
8.1. Concerning the bi-exponential nature of the tropospheric radio
refractive index 311
8.1.1. Introduction and background 311
8.1.2. TV structure in the I.e. A. O. atmosphere 312
8.1.3. Properties of the dry and wet term scale heights 314
8.1.4. Refraction in the bi-exponential model 32 1
8.1.5. Extension to other regions 321
8.2. Effect of atmospheric horizontal inhomogeneity upon ray tracing. _ 322
8.2.1. Introduction and background 322
8.2.2. Canterbury 323
8.2.3. Cape Kennedy 324
8.2.4. Ray bending 326
8.2.5. Comparisons 328
8.2.6. Extension to other regions 331
8.2.7. Conclusions 333
8.3.
8.4.
8.5.
Chapter 9.
9.1.
9.2.
9.3.
9.4.
9.5.
9.6.
Comparison of observed atmospheric radio refraction effects with
values predicted through the use of surface weather observations __
8.3.1. Introduction
8.3.2. Theory
8.3.3. The CRPL standard atmospheric radio refractive
index profile sample
Comparison with independent data
Comparison with experimental results
Discussion of results
of atmospheric refraction errors in radio height finding.
Introduction
Refractive index
Ray theory
Use of the effective earth's radius
Meteorological parameters
Calculation and correlation of height errors
Estimation of the average gradient
Regression analysis
Height error equations
Conclusions
8.3.4
8.3.5
8.3.6
Correction
8.4.1
8.4.2
8.4.3
8.4.4
8.4.5
8.4.6
8.4.7
8.4.8
8.4.9
8.4.10.
References
Radio-meteorological charts, graphs, tables, and sample com-
putations
Sample computations of atmospheric refraction
Tables of refraction variables for the exponential reference
atmosphere
Ref ractivity tables
Climatological data of the refractive index for the United States _ .
Statistical prediction of elevation angle error
References
333
333
334
340
342
344
353
356
356
357
357
360
362
363
363
364
372
373
373
375
375
393
396
404
404
423
Chapter 1. The Radio Refractive
Index of Air
1.1. Introduction
The last few decades have seen a remarkable increase in the practical
utilization of the radio spectrum above 30 Mc/s. This, in turn, has
focussed attention upon the mechanisms by which these radio waves are
propagated. Since radio energy at these frequencies is not normally
reflected by the ionosphere, variability in the characteristics of the re-
ceived fields is attributed to variations in the lower atmosphere and, in
particular, to the radio refractive index.
The radio refractive index is central to all theories of radio propagation
through the lower atmosphere. The atmosphere causes a downward
curvature of horizontally launched radio waves which is normally about
one quarter of that of the earth. Under unusual meteorological con-
ditions, however, the radio energy may be confined to thin layers near
the earth's surface with resultant abnormally high field strengths being
observed beyond the normal radio horizon. At other times a transition
layer between differing air masses will give rise to the reflection of radio
energy. In addition to these gross profile effects, the atmosphere is
always more or less turbulent, with the result that radio energy is scattered
out of the normal antenna pattern.
It is not the purpose of the present discussion to emphasize the inter-
play of various propagation mechanisms, as has been done, for example,
in a classic paper by Saxton [1]', but rather to emphasize that the refrac-
tive index of the troposphere is of central concern in the propagation of
radio waves at frequencies above 30 Mc/s. In what follows, then, the
classical equation for the radio refractive index will be considered, a sum-
mary of the recent determinations of the constants in this equation given,
the errors in the practical use of the equation will be analyzed, and, finally,
the normal gross features of atmospheric refractive index structure will
be described.
1.2. Dielectric Constant of Moist Air
Debye [2] has considered the effect of an impressed electric field upon
the dielectric constant of both non-polar and polar molecules, a polar
1 Figures in brackets indicate the literature references on p. 21.
2 RADIO REFRACTIVE INDEX OF AIR
molecule being one with a strong permanent dipole moment. He derives
the same result from both classical and quantum mechanical methods;
i.e., the polarizability is composed of two effects: one due to the distortion
of all of the molecules by the impressed field and the second arising from
an orientation effect exerted upon polar molecules. The polarization, P,
of a polar liquid under the influence of a high-frequency radio field is
given by
P{o:)
6 - 1 M
£ + 2 p
47riV
ao +
1
3/cr 1 + iojrj
(1.1)
where : e is the dielectric constant,
M is the molecular weight,
p is the density of the liquid,
A^ is Avogadro's number,
ao is the average polarizability of the molecules in the liquid,
assuming no interaction between molecules,
M is the permanent dipole moment,
k is Boltzmann's constant,
T is the absolute temperature,
T is the relaxation time required for external field-induced orien-
tations of the molecules to return to random distribution
after the field is removed,
CO = 2'wf where / is the frequency of the external field.
One concludes from Debye's analysis that for external fields with fre-
quencies less than 100 Gc/s, wr « 1 with the result that (1.1) is written^
€ - 1 M
A^ttN
2
e + 2 p "
3
L""" ^ ^kTA
(1.2)
The dispersive effect of the 22.5 Gc/s water-vapor absorption line is not
expected to be important below 30 Gc/s although, as shall be seen, experi-
mental evidence indicates some dispersion due to the 60 Gc/s absorption
^ The effect of the relaxation time upon interpretation of measurements of dielectric
constant has been discussed by Saxton [1]; he has also given a description of measure-
ment techniques involved in the determination of dielectric constant during the war
years [3].
DIELECTRIC CONSTANT OF MOIST AIR 3
line of oxygen. For non-polar gases (n = 0), this equation becomes
e - 1 M _ 4TNao
6 + 2 p " 3 ^^-"^^
which is well approximated by
e - 1 ^ ^ 4:irNao (1.4)
for gases at low pressures.
This equation may be rewritten, by assuming the perfect gas law, as
e-l =K[^ (1.5)
where K[ is a constant. The result for polar gases, (1.2), may be written
(1.6)
which, assuming (1.5), can be rewritten as
e- 1 ::=K^|(a +|) (1.7)
where K'^, A, and B are constants. For a mixture of gases, Dalton's
law of partial pressures is assumed to hold with the result that we can
sum the effects of polar and non-polar gases and hence obtain
e - 1 =« 1^ 47riV
M
2
"" + SkTA
— 1 = 2] K'li -^ + 22 ^20 i^[Ag-{--~]- (1
•8)
For the troposphere, however, we need only consider the effects of
CO2, dry air (non-polar gases), and water vapor (a polar gas) such that
e - 1 = K,\ :^ + AV, I ( A + I ) + K{, ^ (1.9)
where Pd is the pressure of dry air, c is the partial pressure of water vapor,
and Pc the partial pressure of CO2. The equation for refractive index, n, is
obtained using the expression n = a/m^, where ^i, the permeability, may
4 RADIO REFRACTIVE INDEX OF AIR
be assumed to be approximately unity for air. Since
n = \/l + (/xe - 1) ,
one may employ the approximation
and obtain the familiar form of practical application^
N = (n - 1) 10*' = K, ^ + K, -- + K,jr^-\-K,^ (1.10)
where A'l, A'o, A'3, and A'4 are constants.
1.3. Constants in the Equation for N
A survey of the various deterniinations of Ai, A2, and A3 was recently
made by Smith and Weintraub [4] to arrive at a set of "best values" to
represent the mean of previous independent determinations. In radio
work one is interested in propagation through the troposphere, therefore,
the composition of air should be taken to include an average amount of
carbon dioxide. However, laboratory measurements usually are made
on C02-free air due to variable concentrations of CO2 in the laboratory.
Hence Smith and Weintraub have adjusted the the values of e — 1
originally published for C02-free air by raising them to 0.02 percent to
correspond to a 0.03 percent CO2 content. It should be noted that this
correction of 0.02 percent is essentially of academic interest, since as
shall be seen, the final equation for A'^ will be considered accurate to 0.5
percent. The value of 0.02 percent was obtained by noting that the
value of dielectric constant for C02-free air, e', in the expression
e - 1 = — 2^
could be utilized for applications in the real atmosphere by expressing
total atmospheric pressure, P t, as
Pt = Pd-{- Pc
5 Henceforth, A^ will be used to denote {n — 1)10^ and will not be used again for
Avogadro's number. This slight inconsistency was adopted to maintain notational
agreement with both Debye's early work and the later work in radio meteorology.
CONSTANTS IN EQUATION FOR A^ 5
By assuming a constant COi content of 0.03 percent one obtains, from
Dalton's law of partial pressures,
Pd = (1 - 3 X 10"') Pt
and
P. = 3 X lO-'P,
with the result that the dielectric constant for the real, dry atmosphere,
e, may be written
. K,{\ - 3 X 10~^)P, ^ A' 4(3 X 10~') Pt
€ - 1 = ~ \ .
One may then adjust measurements of C02-free air upwards by
\rTz\ ~ 1 10' = 3 X 10"'(^ - l) percent ^ 0.02 percent,
since Ki/Ki ^^ 5/3. Such values are given in table 1.1 on a real, rather
than an ideal, gas basis. The first determination shown, that of Barrell
[5], is an average of the constant term (n for X == oo ) of the optical Cauchy
dispersion equations for standard air used in three of the principal
measurement laboratories of the world. Theoretical considerations indicate
that the dielectric constant for dry air will be the same for optical and
radio frequencies. Barrell's value is converted to dielectric constant
from the relationship \/m« with ju, the permeability, taken as unity at
optical frequencies. Unless otherwise noted, the standard error is used
here and throughout the remainder of this discussion.
Table 1.1. Dry air refractive index and dielectric constant at 0°C and 1 atm
Reference
Frequency of
measurement
Measured
N
(« - 1) 10«
Year
Barrel! [5]
Optical''
287. 7 ± 0.
575. 6 ± 0. 1
1951
Birnbaum,Kryder,and Lyons[40]
9,000 Mc/s
575.8 ±0.3
1951
Essen and Froome (11] _.
24,000 Mc/s
288. 2 ± 0. 1
576. 1 ± 0. 2
1951
Mean value of {« — 1) 10^
Mean value of N^'
288. 0. ± 0.
575. 7 ± 0. 1
» Maryott and Buckley [6] have determined a mean value for the dielectric constant from the simple
average of eight different determinations that, when adjusted to the pressure, temperature, and value of/x
assumed in table 1.1, yield a value of N = 287. 7 ± 0. 15, which is in agreement with Barren's value.
•> Derived from ;*
meability.
Vm* where m — 1 = 0. 4 X 10"'' is taken for radio froquencies to account for the per-
6 RADIO REFRACTIVE INDEX OF AIR
The statistical mean of the dielectric constant is then converted to
refractive index. The constant Ki is evaluated from
N = Kv
T '
(1.11)
which stems from (1.10) when e = 0. Setting N = 288.04, Pa = 1013.25
mbar, T = 273 °K and solving for K^: then^
Ki = 77.607 ± 0.13
'K
mbar '
(1.12)
The constants K2 and K^ have been evaluated from a survey of deter-
minations of the dielectric constant of water vapor in the microwave
region by Birnbaum and Chatterjee [7]. These values, determined from
the data of table 1.2, are
and
K2 = 71.6 ± 8.5 °K/mbar,
K, = (3.747 ± 0.031) 10'(°K)Vmbar
(1.13)
(1.14)
and were obtained as the weighted means of the various determinations,
the weights being taken inversely proportional to the square of the prob-
able errors. Although Birnbaum and Chatterjee considered only those
data that would allow a determination of Ko and Kz over a wide range
of temperature, they conclude their results are in satisfactory agreement
with those of Stranathan [8] at 1 Mc/s, Phillips [9] at 3000 Mc/s, Grain
[10] at 9000 Mc/s, and Essen and Froome [11] at 24,000 Mc/s.
Table 1.2. Values of the constants K2 and Kz used by Birnbaum and Chatterjee
Observer
A'2
Ki X 10"5 Temperature range
Birnbaum and Chatterjee [7]
69. 43 ± 13. 02
77. 75 ± 21. 70
61. 47 ± 21. 70
72. 86 ± 7. 05
°C
3.774 ± 0.043 25-103
Groves and Sugden [12]
Hurdis and Smyth [13]
3.742 ± 0.097 110-211
3.765 ± 0.096 111-249
Stranathan [8]_ . ._-
3. 736 ± 0. 025 21-189
* The development here follows that of Smith and Weintraub. Grain's [10] de-
termination of the dielectric constant of dry air (e = 1.000572) indicates that Ki may
be somewhat lower than the "best value" of Smith and Weintraub, since his value,
Ki = 77.10, is about 40 standard deviations lower than that of Smith and Weintraub.
CONSTANTS IN EQUATION FOR A^
The full equation for the refractive index is now
^ + 72 I + 3.75 X 10^ jr.
N = 11. Q ^ + 72 ^ + 3.75 X 10' 7^ , (1.15)
after reducing the values to three figures where significant. These are
the constants recommended by Smith and Weintraub to yield an overall
accuracy of ± 0.5 percent in A^.^ Equation (1.15) may be simplified by
assuming Pt = P = Pd + e and obtaining
N = 11.6 I - 5.6 1^ + 3.75 X lO' |^ . (1.16)
For practical work in radio-meteorological studies, (1.16) may be sim-
plified to the two-term expression
I + 3.73 X 10' f.
N = 77.6 ~ + 3.73 X lO' ^ , (1.17)
which yields values of A^^ within 0.02 percent of those obtained by (1.16)
for the temperature range of —50 to 40 °C; i.e., a maximum error of 0.1
A'' unit, and a standard error of 0.5 percent of about 4.5 A'^ units.
This is accomplished by dividing the second and third terms of (1.16)
by e/T and solving for the composite constant, K^, in relation
2:^^-5.6 = 1, (1.18)
which, for T = 273 °K, results in:
^5 = 3.73 X 10'(°KVmbar). (1.19)
Equation (1.17) is commonly Avritten as
iV = ^(p + ^). (1.20)
Although this last equation is in widespread use by radio scientists
throughout the world, it is by no means adopted on an international scale.
^ The values of K2 and Kz given in (1.13) and (1.14) are those of Birnbaum and
Chatterjee as reported by Smith and Weintraub. A consideration of the ratio of
Ko/Ki shows the Smith-Weintraub value to be about 0.2 percent greater than that
obtained from the original data, indicating an arithmetic conversion error. This
error is, however, totally absorbed in the rounding to three significant figures in
(1.15).
8
RADIO REFRACTIVE INDEX OF AIR
Another set of values in common use is that of Essen and Froome (11)
which is listed in table 1.3 along with values determined by several other
authors. Subsequent microwave refractometer determinations of these
constants by Saito [14] in Japan yielded a value of Ki lower than that of
either Smith and Weintraub or of Essen and Froome and nearer to Grain's
determination. Saito's values of Ko and K^ lie between those of Smith
and Weintraub, and Essen and Froome. A recent French microwave
determination of these constants by Battaglie, Boudouris, and Gozzini
[15], has found the Smith and Weintraub values of Ki, Ko, and K^ given
by (1.16) fit their experimental data within measurement error. The use
of the Essen and Froome values in radio geodesy appears to be based
upon Wadley's [16] comparison of optical and radio surveys and his con-
clusion that the best (radio) consistency is obtained when using Essen
and Froome's constants in conjunction with their determination of the
velocity of light.
Table 1.3. Table of constants used by different authors
Pd e e
N = {n - I) 10« = A'l — + K.- + Kz —
Reference
Schelleng, Burrows, and Ferrell [20]
Englund, Crawford, and Mumford [41]
Waynick [35]
Smith- Rose and Stickland [36]
Burrows and Attwood NRDC [37]
Meteorol. Factors in Prop. [40]
Grain [10]
Craig etal. [32]
Essen and Froome [11]
Smith and Weintraub [4]
Essen [38]
Saito [14]
Battaglia, Boudouris, and Gozzini [15].
Magee and Grain [42]
Date
1933
1935
1940
1943
1946
1946
1948
1951
1951
1952
1953
1955
1957
1958
Ki
79.0
79.1
79.0
79.0
79.0
79.0
77.10
79.0
77.64
77.6
77.26
77.6
77.5
Ki
67.5
68.3
68.5
68.0
68.0
68.0
(79)
64.68
72.0
75.0
67.5
72.0
65.0
1.35
3.81
3.72
3.8
3.8
3.8
3.8
3.718
3.75
3.68
3.77
3.75
3.70
The particular constants in the equation for A^ given by Smith and
Weintraub are considered to be good to 0.5 percent in N for frequencies
up to 30,000 Mc/s and normally encountered ranges of pressure, tempera-
ture, and humidity. Experimental determinations of the variability of
the radio refractivity, {n — 1)10^, with frequency have been carried out
by Essen and Froome [11] and are summarized in table 1.4.
Remembering that the dispersion of refractive index would be expected
to be greatest at frequencies slightly off the water vapor resonance at
22 Gc/s and the oxygen resonance at 60 Gc/s [17] one concludes that for
the normally used frequencies, / < 30 Gc/s, the various gases give a
frequency variation of A^ well within the limits of accuracy given by Smith
ERRORS IN USE OF EQUATION FOR A^
Table 1.4. Refractivities of water vapor and dry gases
(Dry gases at °C, 760 mm Hg; water vapor at 20 °C, 10 mm Hg)
Gas
9 Gc/s
24 Gc/s
72 Qc/s
Water vapor
Dry air
Oxygen
60. 7 ± 0. 2
288. 10 ± 0. 10
266. 2 ± 0. 2
60. 7 ± 0. 2
288. 15 ± 0. 10
266. 4 ± 0. 2
61.0 ±0.2
287.66 ±0.11
263. 9 ± 0. 2
and Weintraub. With the exception of oxygen the same is true up to
72 Gc/s. This conclusion is somewhat strengthened by the resuUs of
Hughs and Armstrong [18] at 3000 Mc/s which agree within 1.5 percent
of those of Essen and Froome. It is important to note that the differ-
ences in formulas are greater than the frequency variation and, further,
as shall be seen in the next section, the errors of determining P, T, and e
are sufficiently large to mask even the differences in formulas.
1.4. Errors in the Practical Use of the Equation for N
A high degree of accuracy of temperature, pressure, and water vapor
pressure measurements is necessary for precise determinations of the
refractive index from (1.20). If one assumes that the formula for A^ is
exact, then a relation between small changes in N and small changes in
temperature, pressure, and vapor pressure may be evaluated from
,,, dN -.^ . dN . . dN .j^
(1.21)
assuming that the errors in P, T, and e are unrelated.
The partial derivatives may be evaluated by reference to some standard
atmosphere to yield the approximate expression
AN' = aAT + bAe + cAP.
(1.22)
The root-mean-square (rms) error is then
AN = {(aATY + (bAeY + (cAPy\
2i 1/2
(1.22)
Typical values of the constants a, b, and c, based upon the International
Civil Aviation Organization (ICAO) standard atmosphere and assuming
60 percent relative humidity, are given in table 1.5 for various altitudes.
10 RADIO REFRACTIVE INDEX OF AIR
It was mentioned earlier that the differences in refractive index ob-
tained by the two most commonly used formulas, those of Essen and
Froome and of Smith and Weintraub, are small compared to the error
made in observing P, T, and e. For example, by assuming errors of ±2
mbar in P, ±1 °C in T, and ±5 percent in relative humidity (RH) com-
mon in radiosonde measurements with sea-level values of P = 1013
mbar, T = 15 °C and 60 percent RH, a standard error of 4.1 A^ units may
be obtained compared to the difference of about 0.5 A'^ units between the
values obtained from the two formulas. It is seen from table 1.6 that
(over the normal range of sea-level temperatures, —50 °C to +40 °C)
the differences in the formulas are comparable to those associated with
surface meteorological measurements but are significantly less than the
errors that may arise in radiosonde observations.
In addition to these errors, there is an additional source of error in the
uncertainty of the constants in the equation for N. Equation (1.22)
may be written to include this additional source of error as
AA^' = a^T + 6Ae -f cAP + dAA^ + /AK2 + g^Kz,
and, again assuming that the errors are uncorrected, the rms error may
be evaluated as before. Table 1.7 lists the percentage error arising from
errors in the various surface meteorological observations as well as those
of the constants. It is seen that the errors in constants constitute no
more than 30 percent of the total error. All sources of error combined
will yield errors no larger than 1.5 percent in N. Thus, although the
Smith and Weintraub equation has a stated accuracy of 0.5 percent due to
the constants alone, total errors of nearly twice that figure may occur
under conditions of extremely high humidity.
Currently one must use radiosonde data for estimation of A'^ gradients,
with the result that the overall accuracy is determined much more by the
errors in the meteorological sensors than by errors between constants in
the equation for A''. Until such time as better measurement methods for
T and e are developed, there appears to be little or no need for more
accurate determinations of the constants in (1.10). The different deter-
minations of the constants now available are in essential agreement. For
example, the use of the Essen and Froome expression will give values of
A^ that generally lie within the standard error of ±0.5 percent of the
Smith-Weintraub expression. It a])pears desirable for our pur})oses to use
the Smith-Weintraub constants since they represent the weighted mean
of several independent determinations, noting also that the value of K2
given by Essen and Froome was obtained by extrapolation from optical
measurements rather than direct measurements at radio frequencies.
ERRORS IX USE OF EQUATION FOR A^
11
Table 1.5. Values of the constants a, b, and c in the expression AN = aAT + bAe
+ cAP, for the ICAO Standard Atmosphere and 60 percent relative humidity
Altitude
A^
T
P
e
a
b
c
km
°C
mbar
mbar
°K-i
mbar~'
mbar-^
319
15.0
1013
10.2
-1.27
4.50
0.27
1
277
8.2
893
6.5
-1.09
4.72
0.28
3
216
-4.5
701
2.6
-0.86
5.17
0.29
10
92
-50.3
262
0.04
-0.50
7.52
0.30
20
20
-56.5
55
-0.09
7.96
0.35
50
0.2
9.5
0.8
-0. 0008
4.67
0.27
Table 1.6. Comparison of errors in determining ^ from meteorological measurements
assuming no error in the equation for N
[p = 1013 mbar and RH = 60 percent]
Source of error
-50 °C
0°C
+ 15 °C
+40 °C
Difference in formula.s:
Smith-Weintraub [4] (three term).__
Essen and Froome [11] (three term)..
352. 61
352. 79
306. 18
306. 07
318. 79
318.28
419.55
417.19
Difference from Smith-Weintraub
+0.18
-0.11
-0.51
-2.34
Surface weather observations (±1 mbar of pressure,
±0. 1 °C of T, and ±1 percent RH), (all errors in rnis
A^ units). error
±0.38
±0.43
±0.82
±2.83
Radiosonde observations
of T, and ±5 percent R
(±2 mbar of pressure, ±1 °C
H) (all errors in A" units). rms
error
±1.73
±2.02
±4.07
±14.19
Table 1.7. Percentage contribution of errors in surface meteorological measurements
and constants in the equation for N
Temp (°C)
Percent of total variance due to:
Total rms
AT
Ae
AP
AA'i
AA'2
AK}
error (A'-units)
-50
15
40
16.7
6.2
1.8
0.3
0.0
41.3
64.1
69.3
81.0
35.6
7.9
0.5
2.3
1.0
0,2
0.0
0.0
5.7
10.0
12.7
0.0
10.2
16.0
17.2
0.387
.476
.955
3.378
-50
0.158
0.005
0.348
0.059
0.001
0.002
0.387
.119
.306
.284
.048
.114
.152
.476
15
.127
.765
.269
.045
.302
.382
.955
40
.188
2.811
.248
.040
1.202
.399
3.378
This opinion is particularly reinforced by Essen's subsequent determina-
tion of K2 and A'3 at 9200 Mc/s, where closer agreement was obtained
with the Smith-Weintraub values (note items 9, 10, 11 in table 1.3). The
difference in K2 and /C3 accounts for most of the final difference in A'^ ob-
tained from the two formulas. Although the great precision of Essen and
Froome's work is reflected in their standard deviation of experimental
12 RADIO REFRACTIVE INDEX OF AIR
values being about one-tenth that of other workers, one must recognize
that systematic errors undoubtedly contribute to the disagreement in
values obtained by different workers. It appears that there exists a real
need for a new determination carried out with the greatest possible care
and over as large a pressure and temperature range as possible.
The need for further measurements is particularly evident by the
several Essen and Froome determinations. Their values of the dielectric
constant have been determined with impressive precision and absolute
accuracy over a wide range of frequencies, but their concomitant deter-
minations of the constants in the equation for A'^ differ from one another
as much as the Smith- Weintraub and the 1951 Essen and Froome values.
It is clear that the use of either set of Essen and Froome values permits
the calculation of either relative or absolute values of A'' with equal pre-
cision. However, the question of comparative absolute accuracy with
the results of other workers remains.
As a further example, drawn from radio geodesy, consider the electrical
path length
R = / ndS, (1.23)
Jo
where »S indicates the radio path. If we assume that the measurements
are made in a stratum of constant n, then (1.23) becomes
R = n ds = nS = S{1 -\- N X 10"'),
Jo
and
S = R/{1 -{-NX 10-') c^R{l - N X 10-'). (1.24)
An error in ^V thus directly produces an error in radio distance determina-
tion. In typical radio geodesic field work (see, for example, Wadley,
[16]) A^ would be determined from pressure, dry-bulb temperature, and
wet-bulb temperature readings. The partial pressure of water vapor may
be obtained from Sprung's [19] (psychrometric) formula
e ^ e's - 0.00067 (T - T') P (1.25)
where e', denotes the saturation vapor pressure at the wet-bulb tempera-
ture, r'. For sea-level conditions of P = 1013 mbar, T = 15 °C and
T — T' = 4.1 °C (60 percent RH) one finds, neglecting errors in the
constants, that
AA^' ^ 0.28AP - 4.32A7^ - 6.96Ar. (1.26)
PRESENTATION OF A^ DATA 13
Assuming errors of it 0.1 mbar for P and ±0.25 °C for T and T' (de-
rived from considering Wadley's field data as being accurate to ±5 units
of the next significant figure beyond the tabulated whole degrees Fahren-
heit for wet and dry bulb temperatures and hundredths of an inch of
mercury for total air pressure) one finds that errors in A^ may be as large
as 2.75 N units, which is much greater than the 0.5 N unit difference given
by the two formulas for N. Again it is seen that the difference in the
constants used in the two most widely accepted expressions for N yields
an error of the same size or smaller than that produced by errors of meas-
urement of the necessary meteorological parameters.
1.5. Presentation of N Data
There is now, in the literature, an almost bewildering choice of modi-
fications to basic N data when presented as a function of height. The
underlying principle is always to remove the syste?natic decrease of N with
height, h, in an assuincd standard atmosphere. This arises from the fact that,
the curvature of a radio ray, C, is given by ^
C = - "^rcos0 (1.27)
n dh
where 6 is the local elevation angle of the ray. Since the curvature of a
radio ray is proportional to the gradient of the refractive index, specifica-
tion of a model for dn/dh specifies the curvature of radio rays in that model
atmosphere. For example, it is customary for radio engineers to think in
terms of effective earth radius factors [20]. This convenient fiction,
discussed in chapter 3, makes straight the actual curved path of a radio
ray in the atmosphere by presenting it relative to an imaginary earth
larger in radius by a factor k than the radius of the real earth, a, thus
maintaining the relative curvature between earth and radio ray.
This is expressed as
1 I dn 1
+ - -,7 cos d = y- -\- ,
a n dh ka
curvature of curvature of curvature of curvature of
earth radio ray effective earth straight ray
^Equation (1.27) as well as general ray curvature considerations are elegantly
derived by Millington [21] for the effective earth's radius model. The same equation
is also derived in chapter 3.
14 RADIO REFRACTIVE INDEX OF AIR
Thus k is defined by
1 + iT" COS 6
n an
which, for rays tangential to the earth (60 = 0) and assuming n in unity,
is approximated by
^ - 1 + a dn/dh ' ^^-^^^
It is customary to set
dn/dh = - 7- , (1-30)
and thus obtain k = 4/3. Since l/4a is close to the observed gradients
of n, this represents a "standard atmosphere" gradient and one may re-
move the "standard" decrease of N with height by adding a quantity, 8N
m = {h/4a) 10' (1.31)
to N{h) thus obtaining the "B unit" or
B(h) = [(n - 1) + h/ia] lO' = N(h) + (/i/4a) lO'. (1.32)
This method of presenting refractive index profiles has been shown to
be very useful in emphasizing the departure from standard of the atmos-
phere over southern California [22]. The principal advantage of the
B unit is that any departure from a vertical line, and thus constant n
gradient of dn/dh = — l/4a, equals a departure in refractive bending from
that in the effective earth's radius atmosphere with k = 4/3.
A very similar approach is used to aid in the study of extended ranges
of radio waves. The most pronounced case of this phenomena occurs
when dn/dh = — (1/a) which, from (1.29), gives k = 00 or an effective
earth of infinite radius. This then implies, for radio purposes, the earth is
flat and comnmnication is between "radio-visible" terminals. For this
special study the modified index of refraction, M, is defined by
M{h) = [{n - 1) + h/a] lO' = N(h) + (h/a) lO'. (1.33)
A gradient of dM/dh = implies that A = oc and also implies a theo-
retically infinite range of radio signals. M units were used to present the
PRESENTATION OF N DATA 15
meteorological data gathered by the Canterbury Project [23] in their
intensive study of ranges of over-water radar signals. Both B and M
units assume a standard atmosphere with a linear decrease of A'^ with
height and thus introduce a correction which increases linearly with
height. If the actual atmosphere had in fact the assumed A^ distribution
then, for example, B{h) = B{0). Recent studies [24, 25, 26] have shown
than an exponential decrease of A'^ with height is a more realistic model of
the true atmosphere.
For example, the data given on figure 3.3 were chosen to represent the
extremes of average N profile conditions over the United States. The
Miami, Fla., profile is typical of warm, humid, sea-level stations that
tend to have maximum refraction effects while the Portland, Me., profile
is associated with nearly minimum sea-level refraction conditions. Al-
though Ely, Nev., has a much smaller surface A^ value than either Miami
or Portland, its A^ profile falls within the limits of the maximum and
minimum sea-level profiles. The A'^ distribution for the 4/3 effective
earth's radius atmosphere is also shown on figure 3.3.
It is quite evident that the 4/3 earth distribution has about the correct
slope in the first kilometer above the earth's surface but decreases much
too rapidly above that height. It is also seen that the observed refrac-
tivity distribution is more nearly an exponential function of height than a
linear function as assumed by the effective earth's radius model. The
exponential decrease of N with height is sufficiently regular as to permit a
first approximation of average vV structure from surface conditions alone.
Consider that
N{h) = Nsexp i-h/H), (1.34)
where H iasi scale height appropriate to the value of A'' at zero height, Ns-
Average values of N^ and H for the United States are approximately 313
and 7 km respectively. As used here, scale height is simply that height
at which N{h) is l/e of A''^ under the assumption of (1.34).
One would exj)ect the gradient
^^ = - (Ns/H) exp i-h/H) (1.35)
also to be sufficiently well-behaved as to allow prediction of at least its
general features. In fact a high (;orrelation between AN, the simple
difference between A" at the earth's surface and at 1 km above the earth's
surface, and A^,, has been observed in a number of regions. This is
16 RADIO REFRACTIVE INDEX OF AIR
particularly true of climatological data such as 5-year averages for given
months. In the United States the expression [26]
-AN = 7.32 exp (0.005577iV.), (1.36)
TlnAN . T's = 0.93,
where r denotes the standard correlation coefficient, has been found, while
in Germany [27] the expression
-AN = 9.30 exp (0.004:5Q5Ns), (1.37)
^InAiV • Ns — 0.6,
has been reported. A recent article by Lane [28] indicates that a similar
correlation, rin^" 'Ns= 0.85, is found over the British Isles. Although
the slopes and intercepts of this relation between AA'^ and Ng obviously
vary with climate, one concludes, for the purposes of the present discus-
sion, that the observational evidence corroborates the assumption of an
average exponential decrease of A^ with height. The general application
of the exponential model nmst await considerable work upon the part of
the radio climatologists.
This has naturally led to the definition of a new unit that adds a correc-
tion factor to account for this exponential decrease [29, 30]. This new
unit. A, is given by
A(h) = Nih) + A^.[l - exp {-h/H)] (1.38)
It is i)ossible to calculate a theoretical value of the scale of N by
as.suming a distribution of water vapor. This is, however, quite a
complex procedure. Futhermore, the value obtained depends upon the
model of the water vapor distribution, and no definite conclusion can be
justified considering the extreme variation of water vapor concentration
with season, geographic location, and height above the earth's surface. A
convenient and simple alternative is to adopt a value for H from the
average (n — 1) variation with height in the free atmosphere. Several
such values of H were determined by reference to the National Advisory
Committee for Aeronautics (NACA) standard atmosphere and recent
climatological studies of atmospheric refractive index structure [26]. It
is seen in table 1.8 that H varies from 6.56 to 7.63 km in the NACA
standard atmosphere, depending on the value of the relative humidity
assumed. The value of // = 7.01 km for SO jiercent relative humidity is
PRESENTATION OF N DATA
17
in close agreement with H = 6.95 km obtained from climatological
studies of (n — 1).
Table 1.8. Determination of the effective scale height, H, for the radio refractive index
Source
Humidity //
percent
100
80
60
As observed
6.56
Climatological data
adopted for this discussion.
7.01
7.63
6.95
7.0'
*The value of H = 7. km was arbitrarily adopted for use in the present discus-
sion. Note that this value is smaller than the usual scale height near sea level of
8. km for the density distribution of the air. Although the value H is quite arbi-
trary, it is evident from table 1. 8 that H should lie between 6. 5 and 7. 5 km.
A comparison of B, M, N, and .4 units is shown on figure 1.1 for arctic,
temperate, and tropical climates. It is quite evident that both M and B
over-correct the profile to produce increasing values with height while
the A unit tends to yield a constant value at heights in excess of 2 km.
It is noted, however, that important departures of the Nik) profile from
normal within the first few kilometers are emphasized by both the A and
B profiles.
In radio meteorology, as in other branches of meteorology, it is often
convenient to express the potential value of the refractive index referred
to some standard pressure level. In practice this is found by adding to
the values at any height the product of the iV lapse rate and the height.
This has already been done in the case of the B unit, while the A unit
simply adds the total decrease of N in an exponential atmosphere from
the surface to the height under consideration. Yet another approach is
to replace the values of pressure, temperature, and vapor pressure in the
expression for A^ with their values at some desired pressure level. This
unit, called the potential refractive modulus, (/>, is then defined by
77.(5
Po + 4810 -}
(1.39)
where d is the potential temperature and eo the potential partial pressure
of water vapor, both referred to the reference pressure Po. This formula
is that of Katz [31] with the Smith-Weintraub (constants.
The potential temperature is defined as
d = T{Po/p)'
(1.40)
18
RADIO REFRACTIVE INDEX OF AIR
\
ISACHS
ON, N.V\
/.T, FEBRUARY
1
b/
y
\
s.
V
/
(
N(
\
/
/
k
>
/
A=N(h)H
-B=N(h)i
M=N(h)-(
3i3.o[i-exp.(-^)]
k^
K
M
^
i.
_^
'
\
WA
SHINGT
DN, DC,
MAY
1
B,
y
>
\
A
^
N
{h)\
/
/
\
,
/
>
\
/
-jh-^^
^U
_^_
-^
\
c/
\NTON 1
SLAND,
AUGUST
B-'
X
\
\
A
^
/
N(h\
/
y'
\
/
N
/
M
^^^
5>^
__-
50 100 150 200 250 300 350 400 450 500 550
N
Figure 1.1. Variation of N, A, B, and M profiles with climate.
PRESENTATION OF A^ DATA 19
and eo as
eo = e(Po/p), (1.41)
where R is the universal gas constant, w the molecular weight of air, and
Cjc the specific heat of air at constant pressure. By setting Po = 1000
mbar and noting that R/mCp = 0.286, one obtains
• . = ^(^r + 3.73X.O-|,(l<^r. (1.42)
Thus, it is seen that <^ is obtained by correcting the dry air term of A'^,
77.6p/T, by the factor (lOOO/p)"-^" and the water vapor term by the
factor (1000/p)°-^-'^. In well-mixed air, 6 and Co are independent of height
and conseciuently is independent of height. If the temperature lapse
rate is less than the dry adiabatic rate, the potential temperature increases
with height and, conseciuently, the first term of (/> decreases with height.
If the lapse of the partial pressure of water vapor is less than than given
by (1.41) then eo and the second term of </> increase with height, conversely,
if e decreases more rapidly with height than the decrease given by (1.41),
eo and the second term of decrease with height. Abundant examples of
the vertical distribution of </> under various meteorological conditions are
given throughout volume 13 of the Radiation Laboratory Series [32].
The potential refraction modulus has recently been used by Jehn [33] to
illustrate the variability of A^ about a classic polar-front wave.
A conceptually similar approach to that above is to arbitrarily adopt a
reduced-to-sea-level value of refract ivity
A^o = A^. exp (-H/i/7.0) (1.43)
where h is in kilometers, that effectively removes the station elevation
dependence of N s [34] and allows the emphasis of air-mass differences.
Since the effects of the atmosphere upon the propagation of radio waves
are dependent upon either the absolute value of n or its gradient, we may
use this as a criterion for the choice of appropriate units to represent the
A^ profile. One may recover both n and its gradient from a knowledge of
the height structure alone. The corrections are additive in the case of
B, M , and A and nmltiplicative for A^'o. The values B, M, and A are in
convenient form for refraction calculations since it is the gradient above
the surface of the earth that is needed. The value, A'^o, removes the effect
of station elevation and t hus is convenient for mapping of surface condi-
tions in a fashion similar to sea-level pressure rather than station pressure.
One cannot obtain the gradient and absolute value of n from a 0(/O distri-
bution, although <l>(h) may be graphically calculated from the familiar
20
RADIO REFRACTIVE INDEX OF AIR
characteristic diagram of the meteorologist; even for that process one
must first calculate the potential values of both temperature and water
vapor pressure. The relative merits of the various units are summarized
in table 1.9.
1.6. Conclusions
The above discussion has emphasized the following points:
(a) The differences in constants in the expression for A'' are small com-
pared with the error inherent in the formula.
(b) The refractive index is effectively nondispersive for frequencies
below, say, 50 Gc/s.
(c) The choice of atmospheric model for A'^, and concomitant units,
depends upon the application at hand.
(d) The atmosphere, on the average, yields an exponential d stribution
of A^" wath height.
(e) The use of an exponential model facilitates the preparation of
climatic maps of A'' and makes clear the effect of non-normal A^ structure
upon the bending of radio rays.
Table 1.9. Cotnparison of various units iised in radio meteorology
Unit
B(ft) = .V(/i) + (/i/4a)10i',
M(h)=N(h) + (h/a)W,
N(,{h) = N(h) c\p(h/-.0).
.■nh)=N(h)
+313|l-exp(-/!/7.0)],
.(A)^^X
6/,ooo+^-^»
Oradient
dBjh) _dN{h)
dh " dh
+ (l/4a)106,
dM(h)_dNih)
dh " dh
+ (l/a)10«,
dA'o(ft) dN(h) ( h
dh
dA(h) _ dN(h) 313
dh ~ dh ^ 1
exp(-ft/7.0),
d0(fe) _ d<i>(h) de
dh de dh
d<j>{h) deo.
+
deo dh
Referred
to:
Surface.
Surface.
Sea-level.
Surface.
1000 mbar
Advantages
Vertical distribution
identifies "4/3 earth"
conditions; absolute
value and gradient of n
easily recovered.
Trapping or ducting
layers identified by
{dM/dh)^0, absolute
value and gradient of
n easily recovered.
Removes effects of both
height and station ele-
vation; facilitates map-
ping since N{h) is
easily recovered.
Removes effects of height
n and gradient easily
recovered.
Removes the effects of
height and station ele-
vation, may be de-
rived from character-
istic meteorological
diagram.
Disadvantages
Over-corrects
N(h ) above
the first few
kilometers.
Grossly over-
corrects N{h)
above the
trapping
layer.
Gradient on n
not easily re-
covered.
Does not re-
move the ef-
fect of station
elevation.
Neither n or
dn/dh may
be recovered
from </){ft);
must calcu-
late both
from basic p,
T, and e data
REFERENCES 21
1.7. References
Saxton, J. A. (Sept. 1951), Propagation of metre radio waves beyond the normal
horizon, Proc. lEE 98, 360-369.
Debye, P. (1957), Book, Polar Molecules, pp. 89-90 (Dover Publ. Co., New
York, N.Y.).
Saxton, J. A. (1947), The anomalous dispersion of water vapour at very high
frequencies. Parts I-IV, Book, Meteorological Factors in Radio Wave Propa-
gation, pp. 278-316 (The Physical Society, London, England).
Smith, E. K., and S. Weintraub (Aug. 1953), The constants in the equation for
atmospheric refractive index at radio frequencies, Proc. IRE 41, 1035-1037.
Barrel!, H. (May 1951), The dispersion of air between 2500 A and 6500 A, J. Opt.
Soc. Am. 41, 295-299.
Maryott, A. A., and F. Buckley (1953), Table of dielectric constants and electric
dipole moments of substances in the gaseous state, NBS Circular 537.
Birnbaum, G., and S. K. Chatterjee (Feb. 1952), The dielectric constant of water
vapor in the microwave region, J. App. Phys. 32, 220-223.
Stranathan, J. J. (Sept. 1935), Dielectric constant of water vapor, Phys. Rev. 48,
538-544.
Phillips, W. C. (July 1950), The permittivity of air at a wave-length of 10 centi-
meters, Proc. IRE 38, 786-790.
Crain, C. M. (Sept. 1948), The dielectric constant of several gases at a wave-
length of 3.2 centimeters, Phys. Rev. 74, No. 6, 691-693.
Essen, L., and K. D. Froome (Oct. 1951), The refractive indices and dielectric
constants of air and its principal constituents at 24,000 Mc/s, Proc. Phys.
Soc. (London, England) 64, 862-875.
Groves, L. G., and S. Sugden (1935), Dipole moments of vapors II, J. Chem.
Soc. England 30, 971-974.
Hurdis, E. C, and C. P. Smyth (1942), Dipole moment, induction and resonance
in nitroethane and some chloronitroparaffins, J. Am. Chem. Soc. 64, 28-29.
Saito, S. (Aug. 1955), Measurement at 9,000 Mc of the dielectric constant of air
containing various quantities of water vapor, Proc. IRE 43, No. 8, 1009.
Battaglia, A., G. Boudouris, et A. Gozzini (May 1957), Sur I'indice de refraction
de I'air humide en microondes, Ann. de Telecomm. 12, 181-184.
Wadley, T. L. (1957), The tellurometer system of distance measurement. Empire
Survey Review 14, No. 105, 100-111, and 14, No. 106, 1232-1239.
Born, M., and E. Wolf (1959), Book, Principles of Optics, p. 89 (Pergamon Press,
New York, N.Y., and London, England).
Hughs, J. v., and H. L. Armstrong (May 1952), The dielectric constant of dry
air, J. Appl. Phys. 23, 501-504.
Sprung, A. (1888), Bestimm. d. Luftfeuchtigk. mit Htilfe d. Assmann'schen
Asperationspsychrometers, Das Wetter 5, 105-109.
Shelleng, J. C, C. R. Burrows, and E. B. Ferrell (Mar. 1933), Ultra-short-wave
propagation, Proc. IRE 21, 427-463.
Millington, G. (Jan. 1957), The concept of the equivalent radius of the earth in
tropospheric propagation, Marconi Rev. 20, No. 126, 79-93.
Smyth, J. B., and L. G. Trolese (Nov. 1947), Propagation of radio waves in the
lower atmosphere, Proc. IRE 35, 1198-1202.
Report of P'actual Data from the Canterbury Project (1951), Vols. I-III (Dept.
Sci. Indus. Research, Wellington, New Zealand).
Anderson, L. J. (Apr. 1958), Tropospherit; bending of radio waves. Trans. Am.
Geophys. Union 39, 208-212.
22 RADIO REFRACTIVE INDEX OF AIR
[25] Bauer, J. R., W. C. Mason, and F. A. Wilson (27 Aug. 1958), Radio refraction in
a cool exponential atmosphere. Tech. Rept. No. 186, Lincoln Laboratory,
Massachusetts Institute of Technology, Cambridge, Mass.
[26] Bean, B. R., and G. D. Thayer (May 1959), On models of the atmospheric re-
fractive index, Proc. IRE 47, No. 5, 740-755.
[27] Bean, B. R., L. Fehlhaber, and J. Grosskopf (Jan. 1962), Die Radiometeorologie
und ihre Bedeutung fiir die Ausbreitung der m-, dm,- und cm-Wellen auf
grosse Entfernungen, 15, 9-16, Nachrichtentechnische Zeitschrift.
[28] Lane, J. A. (1961), The radio refractive index gradient over the British Isles,
J. Atmospheric Terrest. Phys. 21, Nos. 2/3, 157-166.
[29] Bean, B. R., and L. P. Riggs (July-Aug. 1959), Synoptic variation of the radio
refractive index, J. Res. NBS 63D (Radio Prop.), No. 1, 91-97.
[30] Bean, B. R., L. P. Riggs, and J. D. Horn (Sept.-Oct. 1959), Synoptic study of the
vertical distribution of the radio refractive index, J. Res. NBS 63D (Radio
Prop.), No. 2, 249-258.
[31] Craig, R. A., I. Katz, R. B. Montgomery, and P. J. Rubenstein (1951), Gradient
of refractive modulus in homogeneous air, potential modulus, Book, Propaga-
tion of Short Radio Waves, ed. D. E. Kerr, pp. 198-199 (McGraw-Hill Book
Co., Inc., New York, N.Y.).
[32] Craig, R. A., I. Katz, R. B. Montgomery, and P. J. Rubenstein (1951), Gradient
of refractive modulus in homogeneous air, potential modulus. Book, Propaga-
tion of Short Radio Waves, ed. D. E. Kerr, p. 189 (McGraw-Hill Book Co.,
Inc., New York, N.Y.).
[33] Jehn, K. H. (June 1960), The use of potential refractive index in synoptic scale
radio meteorology, J. Meteorol. 17, 264.
[34] Bean, B. R., and R. M. Gallet (Oct. 1959), Applications of the molecular re-
fractivity in radio meteorology, J. Geophys. Res. 64, No. 10, 1439-1444.
[35] Waynick, A. H. (Oct. 1940), Experiments on the propagation of ultra-short-wave
Proc. IRE, 28, 468-475.
[36] Smith-Rose, R. L., and A. C. Stickland (Mar. 1943), A study of propagation
over the ultra-short-wave radio link between Guernsey and England on wave-
lengths of 5 and 8 meters, J. lEE 90, No. Ill, 12-24.
[37] Burrows, C. R., and S. S. Atwood (1949), Radio wave propagation, Consolidated
Summary Technical Report of the Committee on Propagation, NDRC, p. 219
(Academic Press, Inc., New York, N.Y.).
[38] Essen, L. (Mar. 1953), The refractive indices of water vapour, air, oxygen,
nitrogen, hydrogen, deuterium, and helium, Proc. Phys. Soc. B 66, 189-193.
[39] Birnbaum, G., S. J. Kryder, and H. Lyons, (Aug. 1951), Microwave measure-
ments of the dielectric properties of gases, J. Appl. Phys. 22, 95-102.
[40] Meteorological Factors in Radio Wave Propagation (1946), (The Physical So-
ciety, London, England) Foreword.
[41] Englund, C. R., A. B. Crawford, and W. W. Mumford (July 1935), Further
results of a study of ultra-short-wave transmission phenomena. Bell Sys.
Tech. J. 13, 369-387.
[42] Magee, J. B., and C. M. Grain (Jan. 1958), Recording microwave hygrometer,
Rev. Sci. Instr. 29, 51-54.
Chapter 2. Measuring the Radio
Refractive Index
2.1. The Measurement of the Radio
Refractive Index
The radio refractive index is defined as the ratio of the speed of propa-
gation of radio energy in a vacuum to the speed in a specified medium.
The radio refractive index may be measured directly if the measuring
instrument is sensitive to the speed of propagation. Refractive index is
measured indirectly by measuring temperature, pressure, and humidity
with subsequent conversion to the refractive index as indicated in the
previous chapter. The direct method employs radio frequency refrac-
tometers to determine the refractive index; the indirect method, standard
weather observations. The direct method appears preferable because
accuracy is dependent upon a single sensor, rather than three and, of
course, because A'^ values are obtained directly. However, refractometers
are relatively complex and expensive devices requiring no small degree of
skill to maintain. Hence, as yet, refractometers are not in general use,
or even available in sufficient quantity to permit large-scale mapping of
refractive index structures. The bulk of the synoptic and climatological
mapping of refractive index is still based on the indirect method of
measurement. The laborious task of converting the measured parame-
ters to refractive index has been somewhat alleviated by the development
of special analog computers [1]' and digital computers.
2.2. Indirect Measurement of the Radio
Refractive Index
The accuracy of the determination of the refractive index from standard
weather observations has been discussed in chapter 1. Figure 2.1 illus-
trates the degree of accuracy to be expected in the measurement of the
refractive index as a function of the accuracies of the sensors for sea-level
conditions. It is assumed that the errors are additive. Planes of equal
' Figures in brackets indicate the literature references on p. 45.
23
24
MEASURING THE RADIO REFRACTIVE INDEX
Figure 2.1. Accuracy of (he delermination of refractivity as a function of the accuracies
of the meteorological sensors.
accuracy are shown for accuracies of ±0.1, ±0.5, ±1.0, and ±2.0 A'^.
It can be seen that a measurement error of ±1.0 A'^ can be contributed
by each sensor if the sensor errors exceed ±0.8 °C for temperature, ±3.7
mbar for total pressure, and ±0.22 mbar for vapor pressure measure-
ments. It is apparent that extreme accuracy is required in the measure-
ment of the water pressure. The temperature can be measured easily to
within several tenths of a degree, the total pressure to within several
millibars, but the humidity is considerably less responsive to the same
degree of relative accuracy. If the wet and dry bulb hygrometer tech-
nique is used to measure the humidity with subsequent conversion using
either the psychrometric formula of Sprung- [2] or List^ [3], the degree of
accuracy is seen to be a function of temperature. For an error of no
greater that ±1.0 A^, the relative humidity must be accurate to within
0.33 percent at 35 °C, while at °C an accuracy of 3.0 percent is necessary.
e = Cs — c , c IS a constant.
566
e = Cs — [fi(l + Co/u.)] p(AT'), Ci and c^ are constants.
INDIRECT MEASUREMENT 25
Experimenters at the National Physical Laboratories [4] have shown that
when an Assnian hygrometer is used with thermometers accurate to
within 0.1 °C, the optimum expected accuracy in the determination of the
water vapor pressure is 0.2 mbar. Errors three times as great can be
expected at extreme conditions. Hence, it would appear that measure-
ment of water vapor limits the accuracy of the determination of the
refractive index to approximately ±1.0 A'^ units.
Temperature, pressure, and humidity are standard measurements of
the world's weather services. Methods of measuring these parameters
are fairly standard. Usually the temperature is read from mercurial or
alcohol thermometers, the pressure from mercurial barometers, and the
huniidity from a conversion of wet- and dry-bulb thermometers. The
degree of accuracy of these measurements is usually a function of the care
exercised by the observer. Thermometers protected by radiation shields
are usually accurate to within ±0.1 °C, barometers to within ±1.0 mbar.
Reading errors can easily be in excess of the instrument error, especially
in the determination of humidity where the wet-bulb depression is subject
to many sources of error. Contamination of the wick or water, insuffi-
cient wetting, and inadequate aspiration are con^mon sources of error.
The wet-bulb determination can be used below freezing, if proper pre-
cautions are observed.
Automatic-recording systems have been devised for measurement of
temperature, pressure, and humidity. The simplest are the hygrother-
mograph and the microbarograph, in which the sensors are connected by
mechanical linkages to pens or chart recorders. The pressure is recorded
by means of an aneroid capsule, the temperature by means of a bimetal
strip or a curved Bourdon tube, and the humidity by means of a hair
hygrometer. The accuracy of such devices limits the determination of
N to within 2 or 3 A^ units.
Sensors producing electrical outputs are used to measure meteorological
parameters in a variety of automatic-recording systems, such as strip-
chart recorders, punched-paper tape, and magnetic-tape recorders. The
more common temperature sensors include resistance thermometers,
thermocouples, and thermistors. The platinum resistance thermometer,
an international standard, is capable of measuring temperature in still
air to within ±0.05 °C if used in a well-compensated bridge circuit.
However, the platinum resistance thermometer is velocity-sensitive in a
moving air stream. Thermocouples avoid this problem and yield short-
term accuracies of approximately ±0.1 °C with time constants measured
in milliseconds.
The total atmospheric pressure can be measured by a variety of elec-
trical sensors. The simplest device is the pressure potentiometer where
an aneroid capsule is mechanically linked to a potentiometer. One or
two millibars can be considered the limit of accuracy in a differential
26 MEASURING THE RADIO REFRACTIVE INDEX
device operating over a range of ±100 nibar with reference to an average
value with time constants of perhaps 5 msec. The capacitive micro-
phone is capable of accuracies of ±0.01 mbar and lag constants of several
milliseconds over a limited range (±2 mbar). The resolution diminishes
to about ±0.1 mbar over the normal range of variations of atmospheric
pressure. The capacitive microphone requires correction for significant
temperature variations. Strain-gage pressure transducers have approxi-
mately the same characteristics as capacitive microphones with some
degradation in lag constants. Differential strain gauge transducers oper-
ating over a range of 4 to 5 mbar may yield accuracies of several hundred-
ths of a millibar and are relatively free of temperature effects.
Relative humidity reciuires the greatest care in measurement. The
common lithium chloride strip is accurate to within ±5 percent in relative
humidity. In general, the lag constant is of the order 8 to 10 sec for
temperatures in excess of °C (see sec. 2.5). Other sensors under de-
velopment show promise. The barium fluoride strip [5] yields accuracies,
hysteresis effects, and response times far superior to lithium chloride but
has the disadvantage of rapid aging. Phosphorous pentoxide sensors [6]
and aluminum oxide sensors [7] appear to be quite promising. The most
accurate method that is available for measuring water vapor pressure is
the wet-dry bulb technique using electrical thermometers (±0.02 mbar).
The lag constant will be relatively large, as it is a function of the wetting
properties of the wick, the rate of aspiration, and even the relative
humidity, and is thus recommended only for temperatures above —24 °C.
Measurements up to several thousand feet can be made from towers or
with tethered balloons or a wiresonde. Only the sensors need be sent
aloft. All auxiliary devices are on the ground, and long cables transmit
the information from the sensors to the recording equipment [8, 9].
The radiosonde is in almost universal use for high-altitude measure-
ment of the meteorological parameters influencing the refractivity. There
are many models of the radiosonde, differing from country to country.
The principle of operation for present American radiosondes is illustrated
in figure 2.2. An aneroid capsule is used as the active element of a
baroswitch. The temperature sensor, humidity sensor, and a reference
resistance are alternately switched into the grid circuit of a blocking
oscillator as the baroswitch wiper moves under the action of decreasing
pressure. The blocking oscillator controls the pulse rate of the rf trans-
mitter; hence, the pulse rate of the transmission is indicative of the value
of the sensor being sampled. The number of the contact energized is a
measure of the pressure; the switching sequence permits identification
of temperature and humidity values. A constant rate of ascent of the
balloon is assumed so that the values of the parameters can be identified
with the proper altitude.
INDIRECT MEASUREMENT
27
RF
TRANSMITTER
BLOCKING
OSCILLATOR
.'
— <
*
(
_,
T
RH
•s
<
<
<
-
:r
EF
EF
?EN
E
CE
ANEROID
CAPSULE
BAROSWITCH
Figure 2.2. Block diagram of the radiosonde.
There is no uniformity of sensors used in the world's radiosondes.
Hence, it is difficult to compare measurements from different countries.
At one international comparison at Payerne, Switzerland, in 1956, 14
nations participated [10]. The results indicated significant differences.
Temperature measurements corresponded to within ±1.5 °C for night
flights but corresponded only to withm ±3.5 °C for daytnne flights.
Pressure measurements agreed well at low altitudes but indicated a dis-
persion of ±1.5 mbar above 9,000 m (29,000 ft), and ±2.5 mbar above
16,000 m (50,000 ft). Humidity comparisons were poor, indicating that
15 percent would be the most optimistic estnnate of the standard devia-
tion from the mean for all flights. On the average, it is estimated that
the standard American temperature sensor indicated a value approxi-
mately 1.5 °C below the mean for all sondes used in the test. The pres-
sure determination was below the mean by 0.5 mbar, whereas the humid-
ity sensor could not be quantitatively evaluated due to the erratic be-
havior of all sensors.
No attempt was made at these comparison trials to determine the
absolute accuracy of any radiosonde. Although the absolute accuracy
of the American sonde has not been determined, satistical evaluation of
28
MEASURING THE RADIO REFRACTIVE INDEX
the uniformity of American radiosondes has been conducted by both the
U. S. Air Force and the U. S. Army Signal Corps. ResuUs indicate a
standard deviation for the temperature sensor of 0.8 °C to 6,000 m and
1.0 °C above that ahitude. The standard deviation for the pressure
determination was 2.2 mbar below 9,000 m and 1.1 mbar above 9,000 m.
Under ideal conditions the humidity sensor (lithium chloride) exhibited
a standard deviation of 5 percent. This accuracy is possible only if the
element is not subjected to high humidity (95 to 100 percent), or satura-
tion by liquid water, and if the temperature is above °C. The response
of the element is especially poor where both temperature and humidity
are low. The measurement means little if the relative humidity is below
15 percent at a temperature of 20 °C, 20 percent at a temperature of
°C, or 30 percent at a temperature of —30 °C.
Hence, "under ideal conditions" at sea level, the standard deviation
in the determination of the refractivity from radiosonde data is approxi-
mately 3 A^ units; at 1 km, the standard deviation would be approxi-
16.7%
33.3%
10
0-
66.7%
;
/
.^
83.
5(
3*.
D%
"i
\y
M"66.7^33.3
S%
50%
66.7%
DECREASING
CHANGE IN RH
100— 50 % -
83.3-33.3%
100-33.3%
< 400
if)
g 300
< 200
16.7%
83
6
"8
5.3-
S.7-
3.3
-100%
/
^
il
H
■^*
i
9>
V
66.
7%
^
"33.3-50%
33.3%
—
66
7-
00%
p
/
/
/
\
y
/
50-
83.3%
y^
Y
«^
'33.3-667%
50%
66.7%
INCREASING
CHANGE IN RH
50-100%
:/t33.3-IOO°/
%
-20 -40 -20 -40 -20 -40 -20 -40
TEMPERATURE (°C)
Figure 2.3. Factors affecting the lag constant of the lithium chloride humidity sensor.
(.\fter Wexler.)
INDIRECT MEASUREMENT
29
TEMPERATURE
SENSOR
TEMPERATURE
HUMIDITY
SENSOR
PRESSURE f
SENSOR
BLOCKING OSCILLATOR
[an /AMT-6(xa-l)l
Figure 2.4. CUnger-Slraiton radiosonde.
(Transmission in N units.)
mately 2 A^ units. This then would appear to be the uhiniate precision
with which present conventional radiosonde sensors can yield the refrac-
tivity.
The lag constants of the radiosonde sensors are also of importance.
Since the sonde is rising at a relatively rapid rate, it passes into regions
of changing refractivity before the sensors are aware of it. The lag
coefficient associated with the radiosonde introduces an error in the esti-
mation of the true gradient. Lag constants of sensors have been ana-
lyzed by Wexler [11] and by Bean and Button [12]. Some correction can
be made to the radiosonde data. Wexler (fig. 2.3) shows that the lag
coefficient of the lithium chloride strip is a function not only of the
temperature, but also of the absolute value of the relative humidity, as
well as of the size and direction of the gradient.
The radiosonde samples temperature and humidity in sequence rather
than simultaneously. Several experimenters have devised means to cor-
rect this deficiency. Misme [13] decreased the cycling time in one radio-
30
MEASURING THE RADIO REFRACTIVE INDEX
L
\ 300 GRAM
RADIOSONDE
/ BALLOON
FREQ a Tyv
Tyv 395 MC L
RECEIVER
395 MC
FREQUENCY
METER
UNIT
_K>.K.
T
RECORDER
ANALOG
COMPUTER
UNIT
T 405 MC 1
T
RECEIVER
405 MC
FREQUENCY
METER
UNIT
|i_i
.KKN^
FREQ. a T
Figure 2.5. Schematic diagram of the Navy Electronics Laboratory (Thiesen) system.
sonde so that many more samples of each parameter were produced per
unit time. CHnger and Straiton [14] developed a radiosonde that com-
bines the parameters so that the output signal is in terms of the refractive
index (see fig. 2.4). Since the wet and dry terms are additive, a parallel
combination of independent conductances can be used. The dry term
sensor has a conductance proportional to temperature and inversely pro-
portional to pressure; the wet term sensor is such that it not only yields
a value proportional to the relative humidity, but also adjusts the relative
magnitude of the two terms. Thiesen [15] devised a similar radiosonde
utilizing two separate sondes on the same balloon (see fig. 2.5). The
temperature and humidity information are combined at the ground sta-
tion by means of an analog computer. For captive balloon application,
Hirao and Akita [8] and Crozier [9] developed similar systems using wet
and dry bulb thermistors to produce direct output in refractive index.
2.3. Direct IVleasurement of the Refractive Index
The resonant frequency of a microwave cavity is a function of its
dimensions and the refractive index of its contents. Hence, if a cavity is
open to the atmosphere, the resonant frequency changes as the refractive
index of the air passing through it changes. If a sealed reference cavity is
DIRECT MEASUREMENT 31
used for comparison, the difference between resonant frequencies becomes
a convenient measure of the refractive index variations in the sampling
cavity. This method of measurement was used in three different instru-
ments: the Grain type, the Birnbaum type, and the Vetter type. The
Grain refractometer utiHzes the cavities as frequency determining ele-
ments in ultra-stable oscillators. The difference in frequency is then the
unit of measure. The Birnbaum refractometer utilizes the cavities as
passive resonance frequencies of the cavities. The time difference between
resonances is the unit of measure. The Vetter instrument, an improve-
ment on Sargent's [16] modification of the Birnbaum's refractometer,
utilizes servo techniques to achieve a null system, thus eliminating the
necessity for extreme electronic stability.
The resonant cavities are significant components in any of these instru-
ments. Of prime importance is the temperature coefficient of the cavity.
Most cavities today are made of invar having a temperature coefficient
of approximately one part per million per degree centigrade [17, 18].
This is equivalent to 1 N unit per degree centigrade. Further tempera-
ture compensation of the cavity has produced temperature coefficients of
0.2 and 0.1 A'^ unit per degree centigrade [19]. Most recently, improved
invar cavities, when compensated, yielded temperature coefficients of
0.03 N units per degree centigrade [20]. The possibility of improving
cavity performance by use of special ceramics [21, 22], with temperature
coefficients (without compensation) of 0.1 N units per degree centigrade
has been investigated.
The response time of the refractometer is a function of how much of
the end plates of the cavity can be opened to the air without appreciable
loss of resonant characteristics. It has been found [23, 24], that as much
as 92 percent of the end plate area could be eliminated without serious
resonant degradation. Such end plates do not impede the flow of air,
and the response time of the instrument is essentially instantaneous.
In general, the accuracy of the refractometer is at least one order of
magnitude smaller than can be achieved by indirect measurement. With
proper care, these instruments are capable of discerning changes of the
refractive index that are less than a tenth of an A'^ unit. As a relative
instrument, i.e., used to measure variations about an undetermined mean,
the time constant is such as to easily allow detection of rates of up to
100 c/s.
The development of microwave refractometers led to an immediate in-
creased interest in the fine structure of refractive index variations and its
application to radio wave propagation. A summary of this development
has been given by Herbstreit [25]. Detailed investigations of the inhomo-
geneities of refractive index structure using spaced cavities yielded signi-
ficant results pertinent to the spectrum of turbulence [26, 27]. Refrac-
tometers have been widely used in aircraft [18, 28, 29]. The Grain
32
MEASURING THE RADIO REFRACTIVE INDEX
refractonieter [29, 30], as illustrated in figure 2.6, utilizes one sealed cavity
as a reference cavity and a ventilated cavity as the sampling element.
Each of these cavities serves as a frequency-determining element of a
Pound [31] oscillator. Very good cavity resonance is required. The
Grain refractonieter is so designed that the cavity resonators operating
in the 9400-Mc/s range are slightly different in resonant frequency. The
resultant difference frequency is the center frequency from which depar-
tures due to changing refractive index are measured. Recent models
utilize a 43 Mc/s center frequency. As the refractive index of the con-
tents of the sampling cavity changes, the difference frequency at the
output of the mixer changes according to
f —
Then, since
AN = An 10^
a change of 1 A^ unit corresponds to a change in the resonant frequency
SAMPLING
CAVITY
REFERENCE
CAVITY
OSCILLATOR
OSCILLATOR
MIXER
AMP
DISCRIW
INATOR
RECO
f\UtR
Figure 2.6. The Crain refractonieter.
DIRECT MEASUREMENT
33
SAMPLING
CAVITY
J — • —
'
PHASE
METER
RECORDER
T
1
REFERENCE
CAVITY
Figure 2.7. The Bimbaum refradomeler.
of the sampling cavity of approximately 9.4 kc/s if the nominal operating
frequency is 9400 Mc/s.
For the measurement of small scale variations of refractive index, a
modified Grain refractometer utilizing a 10.7-Mc/s center frequency with
a 200-kc/s linear range has been used [29]. Scales of 1 to 20 A^ units for
full scale deflections of a 1-mA chart recorder have been used.
The Birnbaum refractometer [32], illustrated in figure 2.7, applies the
resonance principle in a somewhat different fashion. Both the reference,
or sealed, cavity and the sampling, or open, cavity are passive elements.
The cavities are slightly different in frequency and are of the transmission
type having crystal detectors at the output.
A single klystron, frequency modulated by a sawtooth voltage and
having an output frequency linear with time, excites each cavity in se-
quence. During the frequency excursion of the klystron, the respective
cavity resonances are excited, and a pulse formed at each crystal detector.
Since the resonant frequencies of the cavities differ, the two output pulses
will be displaced in time. If the modulation on the klystron is periodic,
then the output pulses will be also displaced in phase, the displacement
being a function of the difference in resonant frequencies. Any change in
the refractive index of the contents of the sampling cavity will alter the
phase difference between the two pulse trains. The relative phase be-
tween the two outputs can then be measured by an electronic phase meter.
34 MEASURING THE RADIO REFRACTIVE INDEX
The phase meter in use is merely a multivibrator having a constant ampli-
tude output. The pulse from each cavity alternately switches the multi-
vibrator "on" and "off." The width of the constant amplitude output
pulse from the multivibrator is a measure of the time difference between
pulses. The resulting train of constant amplitude variable width pulses
is then applied to appropriate recording circuits. The Birnbaum refrac-
tometer is adaptable to multi-cavity operation. A single klystron may
be used to sweep simultaneously a number of spaced sampling cavities.
A single reference cavity permits simultaneous comparison between the
several sampling cavities, and important consideration when the scale and
form of refractive index variations are desired ([27].
The accuracy of the Birnbaum instrument is dependent upon maintain-
ing a linear frequency sweep. This problem can be circumvented in the
manner of Sargent [16], who modified the Birnbaum refractometer to
operate as a microwave hygrometer. A servomechanism is used to tune
the sampling cavity to the resonant frequency of the reference cavity.
The servomechanism positions a tuning probe in the sampling cavity.
The depth of penetration is the measure of the refractive index of the
contents of the sampling cavity. This technique minimizes the depend-
ence on the sweep characteristics of the klystron.
The Vetter refractometer [20], figure 2.8, virtually eliminates the de-
pendence of the refractometer on electronic characteristics and shifts the
limitations to the cavities themselves. In addition, while previous in-
struments are primarily relative refractive index indicators, the Vetter
refractometer was developed as an absolute refractive index device by
using klystron stabilization techniques [33, 34, 35, 36].
Figure 2.8 is a simplified diagram of the Vetter refractometer illustrat-
ing the basic principle of operation. The reference cavity is excited by a
klystron which is modulated by a small sine voltage on the repeller. Any
output at the fundamental modulating frequency at the detector of the
reference cavity is compared in phase to the original modulating signal,
and an error signal is applied to the repeller of the klystron to lock the
center frequency of the klystron to the reference cavity. At coincidence,
the fundamental disappears in the output of the reference cavity. The
same klystron excites the sampling cavity. Any fundamental appearing
at the output after phase comparison to the modulating signal develops
another error signal. This error signal is used to drive a mechanical
servomechanism which tunes the reference cavity to the resonant fre-
quency of the sampling cavity with concomitant shifting of the klystron
center frequency. This then is a double loop system; the reference cavity
controls the klystron; the sampling cavity controls the reference cavity.
Tuning of the reference cavity is accomplished by a motor-driven
probe that penetrates the reference cavity. The design of the probe is
DIRECT MEASUREMENT
35
6>
MODULATING
SINE WAVE
PHASE SENSITIVE
AMPLIFIER
?
ERROR SIGNAL
SAMPLING
CAVITY
SERVO
AMPLIFIER
SERVO
ERROR SIGNAL
READOUT
PHASE SENSITIVE
AMPLIFIER
Figure 2.8. Simplified diagram of the Vetter ref Tactometer.
such as to permit very nearly a linear function of AA'^ versus probe pene-
tration over the range of operation. Once the probe is calibrated with
the cavity, the relative calibration is static and practically independent
of the contents of the cavity. Although the reference cavity is sealed,
there is no necessity for a vacuum. Absolute calibration is accomplished
by determining the zero point, or the intercept, where A'' = 0, by evacuat-
ing the sampling cavity.
Since the Vetter refractometer is mechanically tuned by a servosystem
having considerable inertia, the high-frequency response is relatively poor.
The upper frequency response is 10 c/s. Accuracy and simplicity of
operation are its distinct advantages.
Refractometers have been used to measure both the surface value and
the gradient of the refractive index. Although the units so far described
were designed originally as ground-based equipment, modifications have
36 MEASURING THE RADIO REFRACTIVE INDEX
been made to convert them to airborne use. However, where vertical
gradients to high ahitude are desired, practical aspects of aircraft opera-
tion such as the turning radius and the angle and rate of climb or glide
are factors influencing how well the vertical gradients are estimated.
Considerable horizontal variation in N is often observed. In addition,
the time involved in ascent or descent is relatively long, and the measure-
ments are valid only under static conditions for the total time of measure-
ment. Due to expense and weight, conventional refractometers are not
economical or practical for use in balloon ascent, whether free or tethered.
This led to the development of several lightweight refractometers to be
used either in wiresonde, radiosonde, or dropsonde applications.
Figure 2.9 is a block diagram of a lightweight (6 lb), expendable refrac-
tometer developed by Deam [37] and Deam and Cole [38] for use as a
balloon-borne unit or a dropsonde. The device is a Pound oscillator [31],
operating at nominal frequency of 403 Mc/s; the instantaneous frequency
is determined by coaxial cavity. The refractive index is sampled by the
cavity, and is reflected as a capacitance in the tuned circuit of the oscil-
lator.
Operational tests indicate the electronics are sufficiently stable to pro-
duce the desired accuracy although the cavity size and mass appear to
introduce a sizable time constant. Present accuracy is estimated to be
better than zb5 A'^ units for a complete profile.
The Hay refractometer [39] was developed as a compromise between
the microwave refractometer and the conventional radiosonde (see fig.
2.10). It lacks the accuracy of the microwave refractometer by an order
of magnitude but, weighing only 7 lb, it incorporates the lightweight
feature of the radiosonde. As a balloon-borne instrument it fills a need
for more accurate, faster probing of the refractive index at the higher
elevations.
This refractometer is a 10-Mc/s oscillator whose frequency is deter-
mined by an air-sensing capacitor. A change in the refractive index of
the air passing through the capacitor is reflected as a frequency change
in the oscillator. The sensing capacitor is alternately switched with a
reference capacitor of identical design but which is protected from the
free atmosphere. The use of a reference capacitor diminishes the problem
of temperature compensation since both capacitors will be at very nearly
the same temperature. To further improve the temperature charac-
teristics, the 12 plates of each capacitor are made of invar and are sepa-
rated by 0.25-in. quartz spacers. The output of the 10-Mc/s oscillator
is doubled in frequency and transmitted to the ground station.
DIRECT MEASUREMENT
37
CORRECTION
VOLTAGE.
DETECTOR
HF
OSCILLATOR
AMP
PHASE
DETECTOR
4.5 MC
OSCILLATOR
CRYSTAL
MODULATOR
CAVITY
Figure 2.9. The Deani expendable refradonieter {modified Pound oscillator).
REFERENCE
CAPACITOR
r
- _r
. ^
SWITCH
OSCILLATOR
DOUBLER
AMPLIFIER
^
SENSING
CAPACITOR
Figure 2.10. The Hay refradometer.
38 MEASURING THE RADIO REFRACTIVE INDEX
2.4. Comparison Between the Direct and
Indirect Methods of Measurement
Absolute accuracy may not be the only consideration in the comparison
between the direct and indirect method of measurement. Although
refract ometers may be capable of superior accuracy, the factors of require-
ments, economics, and availability of competent technical personnel may
outweigh this advantage. Refractometers are relatively expensive, some-
what complex, and require competent technical personnel to maintain,
calibrate, and operate them.
In many cases where average values or long-term statistics are adequate,
the use of refractometers may not be indicated. The data on refractive
index structure derived from weather service data has long been used
successfully for the determination of average conditions.
Where extreme accuracy is required, the use of refractometers is indi-
cated. Radar and radio navigation are examples where accurate esti-
mates of both surface values and gradient are necessary to determine the
refraction through the atmosphere. The necessity for true vertical
gradients would demand the use of a balloon-borne or dropsonde type of
refractometer. In many applications the indirect method may be suffi-
cient; however, determination of the fine structure of the refractive index
appears to be presently limited to some type of radio-frequency refrac-
tometer.
2.5. Radiosonde Lag Constants
2.5.1. Introduction
The determination of N from radiosonde data is subject to all of the
errors inherent in the radiosonde observation. Recently Wagner [40]
has analyzed the errors in A^ arising from time lag of the sensing elements,
data transmission techniques, and significant level-selection criteria. Of
these sources of error, Wagner concludes that the time lag of the sensing
elements is the most serious source of error. Further, for the southern
California coastal inversions, Wagner concludes that only the time lag of
the humidity strip need be considered. A similar conclusion has been
reached by Clarke [41] for practical applications involving ship-borne
radar and over-water air-to-air communications. Although there is a
significant correction associated with the time lag of the U.S. radiosonde's
lithium chloride humidity-sensing element, there is also a time lag in the
temperature element, which, as will be shown, must also be taken into
consideration. The correction for the temperature element yields a two-
fold correction to A^ due to the actual error in temperature and the ancil-
lary correction in vapor pressure resulting from the more correct estimate
of the true saturation vapor pressure. This arises from the fact that
RADIOSONDE LAG CONSTANTS 39
when the lithium chloride element measures relative humidity it must be
used with the saturation vapor pressure. Since the saturation vapor
pressure of water is a function of temperature, an error in temperature
produces an error in the estimated water vapor pressure.
The lag constant of the lithium chloride (LiCl) humidity element be-
comes significantly larger for temperatures below °C [42]. Wexler [11]
has made detailed studies of the lag constants of the LiCl elements at low
temperatures under laboratory conditions. Bunker [43], however, has
found quite different lag constants in the free atmosphere. He attributes
this discrepancy in lag constants to the laboratory-determined values
which were obtained for isothermal conditions; whereas, in rising through
the free atmosphere, the radiosonde normally passes from warm to cooler
air. Bunker has raised a serious question, namely that the temperature
lag of the lithium chloride element is possibly as important as the iso-
thermal humidity lags studied by Wexler. It is quite possible that the
interplay of these two lags could produce a total effect either greater or
less than the humidity lag alone. We now have a quandary since there is
not currently in the literature a complete analysis of the interplay of the
temperature and humidity lags of the LiCl humidity element. Although
Bunker considers some aspects of this problem, he does not consider the
case of decreasing humidity and increasing temperature, a typical condi-
tion giving rise to the superrefraction of radio waves. The differences
between Wexler's and Bunker's estimates of the LiCl lag constants are so
great as to make one wonder at the validity of applying any correction for
this effect. Since neither Wexler's nor Bunker's tabulation of lag con-
stants is complete, the choice of lag constant appears to be arbitrary.
Wexler's lag constants will be adopted for the present discussion. Both
the temperature and humidity elements are corrected (as much as is
possible) for their respective time lags in this chai)ter for the purpose of
preparing refractive index profiles.
In what follows the theory of sensor time lag will be examined. Data
from several climatically diverse locations will then be examined to
illustrate the relative importance of the various lag constant corrections
under conditions of superrefraction of radio waves.
2.5.2. Theory of Sensor Time Lags
Middleton and Spilhaus [44] give
f = - -X (^' - "•) (2.1)
as the basic differential equation of the time lag of a meteorological
sensor, measuring the variable 6, where t is time, X the appropriate lag
40 MEASURING THE RADIO REFRACTIVE INDEX
constant, and the subscripts i and e stand for the indicated and environ-
mental values, respectively. The solution to (2.1) depends upon the
manner in which the environmental value of 6 varies. For example, if it
is assumed that de varies linearly with time,
Be -^ e, + ^t, (2.2)
one obtains for the solution to (2.2)
Be - 9, = +^X [1 - exp (-t/\)], (2.3)
as compared to
^ [(/3X - Dexp (t/X)]d,
^ /3X exp m - exp {t/\) ' ^ ^
under the assumption that de varies exponentially with time,
de = 00 exp i-^t). (2.5)
For a column of air, one normally knows the initial reading of the
sensing strij), 0o, and for an assumed linear decrease of 6 with height the
coefficient (8 becomes
rj _ Bj Pq /„ n\
^ ~ t-\[\ - exp (-«/X)]' ^ ^
with the result that (2.2) is written
6e — Qo -{- ', 7T\ 1 . /x M ■ (2-7)
i — X[l — exp ( — V^)J
Once the value of X is determined, estimates of the true properties of the
air can be found at all heights up to the point where the gradient changes.
One may proceed by a different course by noting in (2.1) that
dt ~ dh ' dt ~ ^ dh' ^ ^
where R is ascension rate of the radiosonde (assumed a constant 300
m/min).
If it is further assumed that di varies linearly between reported values
{an assumption compatible with radiosonde reporting procedure), then
R^ = R ^-'"-' ~['\ (2.9)
dh hk+i — hk
where the /cth and the (A- + l)st layers are the boundaries considered.
THEORY OF SENSOR TIME LAGS 41
Thus the environmental value of 6 can be determined from
de,k+i = d.,k+i + RX ^ '^' ~ I'-' , (2.10)
which involves, in a simple fashion, only the indicated or actually meas-
sured values of the parameter d.
One assumes that 6, and Oc are identically the same at /?. = and that
successive application of (2.10) will yield a more realistic estimate of the
distribution of 6 with height. When one applies a correction procedure
of the form of (2.10) to temperature, where the temperature lag constant,
\t, is always 3 sec [45], one obtains immediately the corrected temperature
profile. The same is true for humidity (provided the temperature is
greater than °C), when X/ is assumed to be always 10 sec [40]. Although
there is some indication [46] that, for room temperatures, the humidity
lag constant may be nearer 5 sec. For temperatures less than °C, how-
ever, X/ is a function orf true temperature, true value of relative humidity,
fe, and change of the true value of fe (see fig. 2.3). This means that one
must use an iterative solution for fe since X/ will change as one's estimate
of the true value of /^ and A/^, changes. Since our knowledge of X/ is essen-
tially empirical, the correction procedure is limited to the temperatures
and values of fe and Afe reported by Wexler [11].
In applying the above equations one generally assumes that the time
lags are always known and that the environmental and indicated values
are identical at the base of each layer [47]. These two conditions are
approximately satisfied only for the ground layer since the total lag con-
stant of the humidity strip is not known. It is not clear that any correc-
tion for sensor lag may be made above the initial layer since, for subse-
quent layers, the initial indicated and environmental values are not
identical and, further, lag constants have not been determined for this
case.
2.5.3. Radiosonde Profile Analysis
The utility of the above lag constant corrections is illustrated by
analyzing past radiosonde data for the occurrence of ground-based radio
ducts. A ground-based radio duct is one in which the gradient of N is
sufficient to refract a radio ray to the same curvature as that of the earth.
Thus, for ducting, since ray curvature is given by the gradient of the
refractive index,
4t < ~ = -157 A^ units/km (2.11)
where ro is the earth's radius. The data analyzed were from the months
of expected maximum duct occurrence at Fairbanks, Alaska (Feb.),
42 MEASURING THE RADIO REFRACTIVE INDEX
Washington, D.C. (May), and Swan Island, W.I. (Aug.). All data were
for the year 1953.
As an example of past work, Wagner [40] has assumed that the refrae-
tivity lagged its environmental value according to (2.3) with the result
that
A^, - N, = /3.V X^ [1 - exp {-I/Xm)]. (2.12)
After an analysis of the meteorological conditions of his area of applica-
tion, he set \n = Xf = 10 sec; i.e., the time lag of A^ derived from pres-
sure, temperature, and humidity was identically that of the humidity
sensor for temperature <0 °C. Comparing this method of A'-lag correc-
tion with the uncorrected data (first and last columns of table 2.1), one
notes an increase in both intensity and incidence of ducts in all climates.
This is particularly marked for Fairbanks, where a sixfold incidence in-
crease is obtained. However, when one makes individual time-lag correc-
tions for both temperature and humidity by means of (2.10) a quite
different picture of duct statistics is obtained. For example, the Fair-
banks data indicate a twofold increase rather than a sixfold increase. On
the other hand, Washington shows an incidence of nine rather than six
ducts with a marked increase in N gradient when one corrects for both
temperature and humidity.
The change of ducting statistics at all three locations obtained by the
two methods of time-lag correction yields paradoxical results. The near
contradiction of the two correction procedures is easily explained and
serves as an illustration of the necessity of correcting both the tempera-
ture and humidity elements for general application to ducting statistics.
Consider typical temperature and humidity conditions associated with
ground-based ducts within each climate. Such cases are shown in figure
2.11. It is sufficient to note that the temperate ducts arise from typical
radiation inversion conditions of increasing temperature and decreasing
relative humidity with height; the arctic ducts are associated with the
intense arctic radiation inversion with ground temperature near —25 °C
and nearly constant relative humidity with respect to height; the tropical
ducts, however, appear to be due to slight decreases of both temperature
and humidity with height at temperatures near +25 °C. The effect of
sensor lag upon these different gradients is always to make the indicated
gradient appear less than the true or environmental gradient. Thus
correcting for sensor lags makes the temperature and humidity gradients
more intense. This, in turn, effects the resultant N gradient. One may
write
dN = f~dT + -^ de + jp dP (2.13)
RADIOSONDE PROFILE ANALYSIS
43
Table 2.L Comparison of ground-based ducting statistics derived from various sensor
time lag corrections
Station
No.
profiles
Reported data
Corrected for T and /
sensor lag
Assuming X.v=X/=10
sec and eq (12)
No.
ducts
Average
gradient
No.
ducts
Average
gradient
No.
ducts
Average
gradient
Washington, D.C.,
(May), temperate
climate
Swan Island, (Aug.),
tropical climate. __
Fairbanks, Alaska,
(Feb.), arctic climate..
123
62
51
1
15
2
A'" units/km
-163
-186
-211
9
20
4
N units/km
-202
-206
-212
6
23
12
-181
-238
-250
which, for normal conditions of 15 °C, 1013 mbar, and 60 percent RH, is
approximated by
AAT ^ -1.27 AT + 4.50 Ae + 0.27 AP.
(2.14)
Since, for ducting, a large negative gradient of A^ is required, the effect
of temperature sensor lag correction is to make temperate and arctic
temperature increases wath height more pronounced with a resultant
larger contribution to a negative A^ gradient. Humidity lag corrections
under all conditions lead to a larger decrease of RH with height than
indicated and concomitant more rapid decrease of N with height. Such
an explanation does not seem so evident for tropical conditions, however,
where both temperature and relative humidity decrease with height.
This apparent paradox is explained by the relatively great change in
saturation vapor pressure of water associated with a small change of
temperature near 25 °C which then produces the required large decrease
of e and A'' with height. It is seen, then, that the temperature sensor lag
correction produces an added humidity correction. Thus one must cor-
rect both sensing elements for the purpose of preparing A^ profiles. Even
in the case of the arctic inversion the interplay of the temperature and
humidity lag corrections is very important, as has been noted by Yerg
[48].
Thus, the correction for humidity sensor lag alone tends to overestimate
ducting incidence since the corrected relative humidity decrease, coupled
with the indicated temperature increase, produces a greater decrease in
water vapor pressure than is actually present.
2.5.4. Conclusion
It appears from the present study that if studies of refractive index
profile characteristics are to include sensor lag correction, then allowance
should be made for both temperature humidity sensor time lags, regard-
less of climate. Any systematic application of these conclusions to large
44
MEASURING THE RADIO REFRACTIVE INDEX
bodies of data, however, must await the appearance in the literature of
effective lag coefficients that combine the effects of both temperature and
humidity lags upon the LiCl element.
auw
200
100
r\
300
200
100
SWAN
SLAND
8-2C
)-53
1500 GMT
I
1
1
\
\
\
\
\
Due
WASHINGTON,
D.C.
8-19
-53,
0900
GMT
'i
-
t
Thickness
i 1
1
25 50 75 100 -40 -30 -20 -10 10 20 30 40
Relative Humidity Temperature in Degrees Centigrade
Figure 2.11. Temperalure and humidity profiles.
REFERENCES 45
2.6. References
Johnson, W. E. (Dec. 1953), An analogue computer for the solution of the radio
refractive index equation, J. Res. NBS 51, No. 6, 335-342.
Sprung, A. (1888), Bestimm. d. Luftfeuchtigk. mit Hiilfe d. Assmann'schen
Asperationspsychrometers, Das Wetter 5, 105-109.
List, R. J. (1958), Smithsonian Meteorological Tables, Washington, D.C. (Pub-
lished by the Smithsonian Institution).
National Physical Laboratory, Dept. Science Research (1960), The refractive
index of air for radio waves and microwaves (Teddington, Middlesex, England).
Jones, F. E., and A. Wexler (July 1960), A barium fluoride film hygrometer
element, J. Geophys. Res. 65, No. 7, 2087-2095.
Macready, P. B., Jr. (1960), Field applications of MRI Model 901 water vapor
meter, Internal Memo Report, Meteorol. Research Inc. of Altadena, Calif.
Stover, C. M. (Apr. 1961), Preliminary report on a new aluminum humidity
element. Internal Tech. Memo, Sandia Corporation, Albuquerque, N. Mex.
Hirao, K., and K. I. Akita (Oct. 1957), A new type refractive index variometer,
J. Radio Res. Lab. 4, No. 18, 423-437.
Crozier, A. L. (Apr. 1958), Captive balloon refractovariometer. Rev. Sci. Instr.
29, No. 4, 276-279.
Cline, D. E. (1957), International radiosonde comparison tests. Tech. Memo
NRM-1907, U.S. Army Signal Eng. Lab. Task NR 3-36-11-402.
Wexler, A. (July 1949), Low temperature performance of radiosonde electric
hygrometer elements, J. Res. NBS 43, 49-56.
Bean, B. R., and E. J. Dutton (Nov. 1961), Concerning radiosondes, lag con-
stants, and refractive index profiles, J. Geophys. Res. 66, No. 11, 3717-3722.
Misme, P. (Jan. 1956), Methode de mesure thermodynamique de I'indioe de
refraction de I'air — description de la radiosonde MDI, Ann. Telecommun. 11,
No. 1, 81-84.
Clinger, A. H., and A. W. Straiton (May 1960), Adaptation of the radiosonde for
direct measurement of radio refractive index. Bull. Am. Meteorol. Soc. 41,
No. 5, 250-252.
Thiesen, J. F. (Apr. 1961), Direct measurement of the refractive index by radio-
sonde, Bull. Am. Meteorol. Soc. 42, No. 4, 282.
Sargent, J. A. (May 1959), Recording microwave hygrometer, Rev. Sci. Instr.
30, 345-355.
Lement, B. S., C. S. Roberts, and B. L. Averbach (Mar. 1951), Determination of
small thermal expansion coefficients by a micrometric dilatometer method.
Rev. Sci. Instr. 22, No. 3, 194-196.
Bussey, H. E., and G. Birnbaum (Sept.-Oct. 1953), Measurement of variation in
atmospheric refractive index with an airborne microwave refractometer, J. Res.
NBS 51, No. 4, 171-178.
Grain, C. M., and C. E. Williams (Aug. 1957), Method of obtaining pressure and
temperature in sensitive microwave cavity resonators. Rev. Sci. Instr. 28,
No. 8, 620-623.
Vetter, M. J., and M. C. Thompson, Jr. (June 1962), An absolute microwave
refractometer, Rev. Sci. Instr. 33, 65()-660.
Thompson, M. C, Jr., F. E. Freethey, and D. M. Waters (Oct. 1958), Fabrication
techniques for ceramic X-band cavity resonators, Rev. Sci. Instr. 29, No. 10,
865-868.
Eraser, D. W., and E. G. Holmes (1953), Precision frequency control techniques,
Unal Report Proj., 229-298 (Georgia List. Tech., State Eng. Exp. Sta.).
46 MEASURING THE RADIO REFRACTIVE INDEX
Adey, A. W. (1957), Microwave refractometer cavity design, Can. J. Tech. 34,
519-521.
Thompson, M. C, Jr., F. E. Freethey, and D. M. Waters (July 1959), End plate
modification of A'-band TE cavity resonators, IRE Trans. Microwave Theory
Tech. MTT-7, No. 3, 388-389.
Herbstreit, J. W. (July 1960), Radio refractometry, NBS Tech. Note 66.
Birnbaum, G. (Apr. 1951), Fluctuations in the refractive index of the atmosphere
at microwave frequencies, Phys. Rev. 82, 110-111.
Birnbaum, G., and H. E. Bussey (Oct. 1955), Amplitude, scale, and spectrum of
refractive index inhomogeneities in the first 125 meters of the atmosphere,
Proc. IRE 43, 1412-1418.
Grain, C. M., and A. P. Deam (Aug. 1951), Measurement of Tropospheric Index
of Refraction Profiles with an Airplane-Carried Direct Reading Refractometer,
Electrical Engineering Research Laboratory, University of Texas, Austin, Tex.
Grain, C. M. (Oct. 1955), Survey of airborne microwave refractometer measure-
ments, Proc. IRE 43, No. 10, 1405-1411.
Grain, G. M. (May 1950), Apparatus for recording fluctuations in the refractive
index of the atmosphere at 3.2 centimeter wavelength. Rev. Sci. Instr. 21,
No. 5, 456-457.
Pound, R. V. (Nov. 1946), Electronic frequency stabilization of microwave
oscillators, Rev. Sci. Instr. 17, No. 11, 490-505.
Birnbaum, G. (Feb. 1950), A recording microwave refractometer, Rev. Sci.
Instr. 21, No. 2, 164-176.
Murray, G., and T. B. Watkins (Apr. 1957), Automatic frequency control, U.S.
Patent 2,788,445. Filed Feb. 5, 1952, granted Apr. 9, 1957.
Silsbee, R. H. (Jan. 1956), A high sensitivity paramagnetic resonance spectrom-
eter, Gruft Lab. Tech. Report 221, Harvard Univ., Cambridge, Mass.
Jung, P. (Oct. 1960), Transistorized frequency stabilization for reflex klystrons
used in magnetic resonance, J. Sci. Instr. 37, 372-374.
Smith, M. J. A. (Oct. 1960), Frequency stabihzation of klystrons, J. Sci. Instr. 37,
398-399.
Deam, A. P. (May 1959), An expendable atmospheric radio refractometer, EERL
Report 108, Univ. of Texas, Austin, Tex.
Deam, A. P., and C. F. Cole, Jr. (Jan. 1960), Development of a lightweight
expendable microwave refractometer, EERL Report 5-46, Univ. of Texas,
Austin, Tex.
Hay, D. R., H. C. Martin, and H. E. Turner (June 1961), Lightweight refrac-
tometer. Rev. Sci. Instr. 32, No. 11, 693-697.
Wagner, N. K. (July 1960), An analysis of radiosonde effects on measured fre-
quency of occurrence of ducting layers, J. Geophys. Res. 65, 2077-2085.
Clarke, L. G. (1960), Theory of atmospheric refraction. Part II of Meteorological
Aspects of Radio-Radar Propagation, pp. 31-82, NWRF3 1-0660-035 (U.S.
Navy Weather Research Facility, Norfolk, Va.).
Dunmore, F. W. (1938), An electric hygrometer and its application to radio
meteorography, J. Res. NBS 20, 723-744.
Bunker, A. F. (1953), On the determination of moisture gradients from radiosonde
records, Bull. Am. Meteorol. Soc. 34, 406-409.
Middleton, W. E. K., and A. F. Spilhaus (1953), Book, Meteorological Instru-
ments, p. 63 (Univ. of Toronto Press, Toronto, Canada).
Sion, E. E. (Jan. 1955), Time constants of radiosonde thermistors. Bull. Am.
Meteorol. Soc. 36, 16-21.
Wexler, A., S. Garfinkel, F. Jones, S. Hasegawa, and A. Krinsky (Aug. 1955),
A fast responding electric hygrometer, J. Res. NBS 55, 71-78.
REFERENCES 47
[47] Wagner, N. K. (May 1961), The effect of time constant of radiosonde sensors on
the measurement of temperature and humidity discontinuities in the atmos-
phere, Bull. Am. Meteorol. Soc. 42, 317-321.
[48] Yerg, D. G. (May 1950), The importance of water vapor in microwave propaga-
tion at temperatures below freezing, Bull. Am. Meteorol. Soc. 31, 175-177.
Chapter 3. Tropospheric Refraction
3.1. Introduction
If a radio ray is propagated in free space, where there is no atmosphere,
the path followed by the ray is a straight line. However, a ray that is
propagated through the earth's atmosphere encounters variations in
atmospheric refractive index along its trajectory that cause the ray path
to become curved. The geometry of this situation is shown in figure 3.1,
which defines the variables of interest. The total angular refraction of
the ray path between two points is designated by the Greek letter r, and
is commonly called the "bending" of the ray. The atmospheric radio
refractive index, n, always has values slightly greater than unity near the
earth's surface (e.g., 1.0003), and approaches unity with increasing height.
Thus ray paths usually have a curvature that is concave downward, as
shown in figure 3.1. For this reason, downward bending is usually defined
as being positive.
If it is assumed that the refractive index is a function only of height
above the surface of a smooth, spherical earth (i.e., it is assumed that the
refractive index structure is horizontally homogeneous), then the path of
a radio ray will obey Snell's law for polar coordinates:
riir^ cos 6-2 = niVi cos 9i, (3.1)
the geometry and variables used with this equation are shown in figures
3.1 and 3.19. With this assumption r may be obtained from the following
integral,
Ti 2 = - / cot 5 — , (3.2)
which can be derived as shown later in the chapter or as derived by Smart
The elevation angle error, e, is an important quantity to the radar
engineer since it is a measure of the difference between the apparent
Figures in V:)rackets indicate the literature references on p. 87.
49
50
TROPOSPHERIC REFRACTION
Figure 3.1. Geometry of the refraction of radio waves.
elevation angle, ^o, to a target, as indicated by radar, and the true eleva-
tion angle. Under the same assumptions made previously, e is given as a
function of r, n, and by
e = Arctan
cos T — sin T(tan 6) —
tan ^0 — sin t — cos r tan
(3.3)
INTRODUCTION
51
Figure 3.2. Differential geometry used in the derivation of the effective-earth' s-radius-
model atmosphere.
The apparent range to a target, Re, as indicated by a radar, is defined as
an integrated function of n along the ray path,
r
Jo
Re = I ndR =
ndh
sin d
(3.4)
However, the maximum range error {R^ minus the true range) likely to
be encountered is only about 200 m, hence the evaluation of (3.4) is not
of great importance unless one is dealing with an interferometer or phase-
measuring system.
62 TROPOSPHERIC REFRACTION
The integral for r, (3.2), cannot be evaluated directly without a knowl-
edge of the behavior of n as a function of height. Consequently, the
approach of the many workers in this field has been along two distinct
lines: the use of numerical integration technicjues and approximation
methods to evaluate r without full knowledge of 7i as a function of height,
and the construction of mode) ?i-atmospheres in order to evaluate average
atmospheric refraction. The following sections are devoted to a discus-
sion of these methods.
3.2. Limitations to Radio Ray Tracing
The user should keep in mind that the equations given in the preceding
section are subject to the following restrictions of ray tracing:
(1) The refractive index should not change appreciably in a wavelength.
(2) The fractional change in the spacing between neighboring rays
(initially parallel) must be small in a wavelength.
Condition (1) will be violated if there is a discontinuity in the refractive
index (which will not occur in nature), or if the gradient of refractive
index, dn/dr, is very large, in which case condition (2) will also be violated.
Condition (1) should be satisfied if
(dn /dh) per km
T^ < (J.VVZjkc,
where refract ivity, A^, is defined as .V = (n — 1) X 10^ and/kc is the carrier
frequency in kilocycles per second [2]. Condition (2) is a basic require-
ment resulting from Fermat's principle for geometrical optics. An atmos-
pheric condition for which both conditions (1) and (2) are violated is
known as "trapping" of a ray, and it can occur whenever a layer exists
with a vertical decrease of A'' greater than 157 A'^-units per kilometer. A
layer of this type is called a "duct," and the mode of propagation through
such a layer is similar to that of a waveguide [3]. Taking into account
refractive index gradients, a cutoff frequency may be derived for wave-
guidelike propagation through a ducting layer [4].
In addition to the above limitations, it should be remembered that the
postulate of horizontal homogeneity, made in order to use (3.1), is not
realized under actual atmospheric conditions; some degree of horizontal
inhomogeneity is always ])resent (see chapter 8).
HIGH INTIAL ELEVATION ANGLES
53
3.3. An Approximation for High Initial
Elevation Angles
A method may be derived for determining ray bending from a knowl-
edge only of n at the end points of the ray path, if it is assumed that the
initial elevation angle is large. Equation (3.2) in terms of refractivity,
N, is equal to
Ti,2 = - / cot d dN ■ 10"
(3.5)
assuming w = 1 in the denominator. Integration by parts yields:
Tl,2 =
Ni.e.,
cot 8 dN ■ 10"
A^,,e,
N coid ■ 10"
JA^i.fli
i.A^j
Bx.Ni
N
sin^0
dd ■ 10'
(3.6)
Note that the ratio, N/s,m^d, becomes smaller with increasing 6 for values
of 6 close to 90°. If point 1 is taken at the surface, then di = ^o and
A^i = A^,. Then for ^o = 10°, No = and do = 7r/2, the last term of
(3.6) amounts to only 3.5 percent of the entire equation, and for the same
values of ^"2 and 62, but with ^0 = 87 mrad (-^5°), the second term of (3.6)
is still relatively small (--^10 percent). Thus it would seem reasonable
to assume that for
do > 87 mrad (-^5°),
the bending, n ,2, between the surface and any point, r, is given sufficiently
well by
or
Tl,2 = —
A cot X 10"
^r.»r
-I AT ,,.9(1
Tl,2 = Ns cot do X 10-"^ - Nr cot Or X IQ-^
(3.8)
The term —Nr cot dr X 10~^ is practically constant and small with re-
spect to the first term, for a given value of do and r, in the range ^0 > 87
mrad. Thus ti,2 is seen to be essentially a linear function of N s in the
range do > 87 mrad. For bending through the entire atmosphere (to a
point where Nr = 0), and for do < 87 mrad, (3.8) reduces to
T = N, cot ^0 X 10-
(3.9)
54 TROPOSPHERIC REFRACTION
For initial elevation angles less than about 5°, the errors inherent in this
method exceed 10 percent (except near the surface) and rise quite rapidly
with decreasing ^o-
3.4. The Statistical Method
Another method for determining high-angle bending is the statistical
linear regression technique developed by Bean, Gaboon, and Thayer [5].
It has been found that for normal conditions and all heights the righthand
integral of (3.6) is approximately a linear function of A''^ (^o, r constant)
for 00 < 17 mrad ('~1°) and that the second term of (3.8) tends to be
constant. Thus (3.6) reduces to a linear equation,
ri,2 = hNs + a, (3.10)
where h and a are constants (as in tables 9.1 to 9.9) and N s is the surface
refractivity.
The form of (3.10) is very attractive, since it implies two things:
(1) The value of n ,2 may be predicted with some accuracy as a function
only of N s (surface height and ^o constant), a parameter which may be
observed from simple surface measurements of the common meteorological
elements of temperature, pressure, and humidity.
(2) The simple linear form of the equation indicates that, given a
large number of observed ri,2 versus N s values for many values of h and
da, the expected (or best estimate) values of h and a can be obtained by
the standard method of statistical linear regression.
This is what was done to obtain values listed in tables 9.1 to 9.9.
Tables 9.1 to 9.9 also show the values of the standard error of estimate,
SE, to be expected in predicting the bending, and the correlation coeffi-
cients, r, for the data used in predicting the lines. Linear interpolation
can be used between the heights given to obtain a particular height that
is not listed in the tables. For more accurate results, plot the values of
r from the tables (for desired N s) against height, and then plot the values
of the standard error of estimate on the same graph. Then connect these
points with a smooth curve. This will permit one to read the r value and
the SE value directly for a given height.
3.5. Schulkin's Method
Schulkin has presented a relatively simple, numerical integration
method of calculating bending for A-pro files obtained from ordinary
significant-level radiosonde (or "RAOB") data [6]. The A profile ob-
tained from the RAOB data consists of a series of values of A for different
SCHULKIN'S METHOD 55
heights; one then assigns to N(h) a Hnear variation with height in between
the tabulated profile points, so that the resulting A^ versus height profile
is that of a series of interconnected linear segments. Under this assump-
tion, (3.2) is integrable over each separate linear A'^-segment of the profile
(after dropping the n term in the denominator, which can result in an
error of no more than 0.04 percent in the result), yielding the following
result :
An.2(rad) ^ - f"'"' cot d dn ^ '^^'^' ~ ""'^
tan di + tan 62 '
or
ATi.2(mrad) ^ — „ . . — . (3.11)
tan 0] + tan d-i
For the conditions stated above, this result is accurate to within 0.04
percent or better of the true value of An, 2, an accuracy that is usually
better than necessary. Thus it is possible to simplify (3.11) further by
substituting 6 for tan 6; this introduces an additional error that is less than
1 percent if 6 is under 10° (^^175 mrad). Now (3.11) becomes
2 (A/", — N'>)
Ari,2(mrad) ^ - q^ j^ q^ , (3-12)
(mrad) (mrad)
where 6 may be determined from (3.58).
The bending for the whole profile can now be obtained by summing
up the An, 2 for each pair of profile levels:
r.(mrad) S t ^ ^f "f"^'' . (3.13)
(mrad) (mrad)
This is Schulkin's result. The degi'ee of approximation of (3.13) is
quite high, and thus most recent "improved" methods of calculating r will
reduce to Schulkin's result for the accuracy obtainable from RAOB or
other similar data. Thus, provided that the A'^-profile is known, (3.13) is
the most useful form for computing bending (for all practical purposes)
that should concern the communications or radar engineer. Some other
methods have been published which are actually the same as Schulkin's,
but have some additional desirable features; e.g., the method of Anderson
[7] employs a graphical approach to avoid the extraction of s(iuare roots
to obtain 0^.
56 TROPOSPHERIC REFRACTION
3.6. Linear or Effective Earth's Radius Model
The classical method of accounting for the effects of atmospheric
refraction of radio waves is to assume an effective earth's radius, a^ = ka,
where a is the true radius of the earth and k is the effective earth's radius
factor. This method, advanced by Schelleng, Burrows, and Ferrell [8],
assumes an earth suitably larger than the actual earth so that the curva-
ture of the radio ray may be absorbed in the curvature of the effective
earth so that the relative curvature of the two remains the same, thus
allowing that radio rays be drawn as straight lines over this earth rather
than curved rays over the true earth. This method of accounting for
atmospheric refraction permits a tremendous simplification in the many
practical problems of radio propagation engineering although the height
distribution of refractive index implied by this method is not a very
realistic representation of the average refractive index structure of the
atmosphere. This section will consider the refractive index structure
assumed by the effective earth's radius model and how this differs from
the observed refractive index structure of the atmosphere. Further, the
limits of applicability of the effective earth's radius approach will be ex-
plored and a physically more realistic model, the exponential, will be
described for those conditions where the effective earth's radius model is
most in error.
It is instructive to give a derivation of the expression relating the curva-
ture of radio rays to the gradient of refractive index. In figure 3.2 a
wave front moves from AB to A'B' along the ray path. If the phase
velocity along AA'isv and v + dv along BB', then, from considering the
angular velocity,
'- = '-^ (3.14)
p p -j- dp
or
^ = ^ (3.15)
V p
where p is the radius of curvature of the arc A A'. Now, since the phase
velocity, v, is
V = - (3.16)
n
where c is the velocity of light in vacuo, one obtains
dv^_dn (3^^)
V n
EFFECTIVE EARTH'S RADIUS 57
combining (3.15) and (3.17), the familiar equation,
i=-^f^, (3.18)
p n dp
is obtained. If the ray path makes an angle 6 with the surface of constant
refractive index
dr = dp cos d (3.19)
and
1 = _ i ^ cos e. (3.20)
p n dr
If the curvature of the effective earth is defined as
tte a p
then
(3.21)
and
Ue = ka ^ .. ^ .. (3.22)
1/a — 1/p
k = ^ . (3.23)
a dn
1 + — 77 cos d
n dn
For the small values of 6 normally used in tropospheric propagation, cos 6
may be set equal to unity. Further, by setting
I - - fa ^'-'^^
one obtains the familiar value of k = 4/3 for the effective earth's radius
factor. By assuming that the gradient of 7i is constant, a linear model
of N versus height has been adopted.
For this model, the bending
ri.2 = — / cot d dn (3.25)
is written
-'=£'1^'* (3.26)
58
TROPOSPHERIC REFRACTION
since
and
N = No - ^ 10
4a
dn = dN X 10-' = - ?
4a
(3.27)
(3.28)
Further, for the case hi = ho ^ 0, and
< 00 < 10°,
where do is the initial elevation angle of a ray, (3.26) may be approximated
by
TO,h —
dli
4ad '
The angle 6 may be determined from
7. - )dl-h 2{N - No) +l(h- ho) • 10'
1/2
u-iir
For the case when ^o = 0, (3.29) becomes
1 I" dh
TO.h
2\/6a '^'^ Vh VQ
1
To.h —
Vq
1 Ih
Vh/a.
Now, from the geometrical relationship,
To.h = ' h (do ~- Oh),
(3.29)
(3.30)
(3.31)
(3.32)
(3.33)
(3.34)
one finds, for do = 0,
do.h ~ a(ro,/i + dh),
(3.35)
MODIFIED EFFECTIVE EARTH'S RADIUS 59
which upon substitution from (3.33) and (3.31) gives
do.ft = V2/i(4/3) a, (3.36)
or, more familiarly,
do,h = V2kah . (3.37)
A very convenient working formula is derived from (3.37) hy k = 4/3,
a = 3960 miles and using units of miles for the ground distance to the radio
horizon, do,h, and feet for the antenna height, h:
do.h = V2h miles (3.38)
This is the familiar expression often used in radio propagation engineer-
ing for the distance to the radio horizon.
3.7. Modified Effective Earth's Radius Model
The effective earth's radius model, although very useful for engineering
practice, is not a very good representation of actual atmospheric A^ struc-
ture. For example, the data on figure 3.3 represent the average of
individual radiosonde observations over a 5-yr period at several locations
chosen to represent the extremes of refractive index profile conditions
within the United States. The Miami, Fla., profile is typical of warm,
humid sea-level stations that tend to have maximum refraction effects
while the Portland, Me., profile is associated with nearly minimum sea-
level refraction conditions. Although Ely, Nev., has a much smaller
surface A'^ value than either Miami or Portland, it is significant that when
its A'^ profile is plotted in terms of altitude above sea level, it falls within
the limits of the maximum and minimum sea level profiles. It is to take
advantage of this simplification, that altitude above sea level rather than
height above ground is frequently used throughout this monograph. The
A'' distribution for the 4/3 earth atmosphere is also shown on figure 3.3.
It is quite evident that the 4/3 earth distribution has about the correct
slope in the first kilometer above the earth's surface but decreases much
too rapidly above that height. It is also seen, by noting that figure 3.3 is
plotted on semi-logarithmic paper, that the observed refractivity distri-
bution is more nearly an exponential function of height than a linear
function of height as assumed by the 4/3 earth atmosphere. One might
expect the refractivity to decrease exponentially with height since the
first term of the refractivity equation (1.20) involving P/T, comprises at
least 70 percent of the total and is proportional to air density, a well-
known exponential function of height.
60
TROPOSPHERIC REFRACTION
N
40
N
^
Miami, August, 15:00 G.M.T.
^Portland, Me., February, 15.00 G.M.T
\
^
y, i^sL
bruary
, 15:00 GM.T
\
\
rApproximate data limits for
\ individual radiosonde profiles
Y^^^^ r Maximum
\ ^^^^ rMean
\ ^^^>/ yMlnimum
^
*
\
■
/3 ea
rth pre
ifiles
2 4 6 8 10
Altitude Above Mean Sea Level in Kilometers
Figure 3.3. Typical N versus height distrihulions.
One might wonder, in the light of the data of figure 3.3, why the effective
earth's radius approach has served so well for so many years. It appears
that this success is due to the 4/3 earth model being in essential agree-
ment with the average A^ structure near the earth's surface which largely
controls the refraction of radio rays at the small values of 6^ common in
tropospheric communications systems.
It would seem that the deficiency of the effective earth's radius ap-
proach could be lessened by modifying that theory in the light of the
average A^ structure of the atmosphere. An indication of the average A^
structure was obtained by examining a variety of N profiles which were
carefully selected from 39 station-years of individual radiosonde obser-
vations to represent the range of A'^ profile conditions during summer and
winter at 13 climatically diverse locations. The results of this examina-
tion are given in table 3.1.
MODIFIED EFFECTIVE EARTH'S RADIUS
61
Table 3.1.
Refradivity statistics as a function of altitude above sea level as derived
from individual radiosonde observations
Altitude
N
Maximum A^
Minimum A'
Range*
{km)
4.
197.1
209.5
186. 5
23.0
5.
172.3
151.4
184.0
161.0
165.0
146.0
19.0
6.
15.0
7.
134.0
139.5
129.5
10.0
8
118.4
104.8
121.5
108.0
113.3
100.0
8.2
9_
8.0
10.
92.4
97.0
86.0
11.0
11.
81.2
86.0
70.0
16.
12.
70.7
76.0
60.5
15,5
14.
53.2
60.0
44.5
15.5
•Range = maximum A^ — minimum A^.
It is interesting to note that the range of A^ values has a minimum at
8 to 9 km above sea level but is systematically greater above and below
that altitude. The average value of 104.8 at 9 km corresponds to a
similar value reported by Stickland [9] as typical of the United Kingdom.
Further the altitude of 8 km corresponds to the altitude reported by
Humphreys [10] where the atmospheric density is nearly constant regard-
less of season or geographical location. Since the first term in the expres-
sion for refractivity is proportional to air density, and the water-vapor
term is negligible at an altitude of 9 km, the refractivity also tends to be
constant at this altitude. It seems quite reasonable, then, to adopt a
constant value of A^ = 105 for 9 km, thus further facilitating the specifica-
tion of model atmospheres. Further, as also noted in chai)ter 1, when
the values of table 3.1 are plotted such as on figure 3.3, it is seen that the
data strongly suggest that N may be represented by an exiwnential
function of height of the form:
N{h) = iVoexp {-bh},
in the altitude range of 1 to 9 km above .sea level.
The following recommendation is made when dealing with problems
involving ground-to-ground communications systems or other types of
low-altitude radio propagation problems where the ray paths involved do
not exceed 1, or at most 2, km above the earth's surface: use the effective
earth's radius method to solve the associated refraction problems. The
u.ser should refer to the tables in chapter 9, where effective earth's radius
factors are tabulated along with other refractivity variables. Table 9.27
may be entered with N ^ and table 9.28 may be entered with AN{N s sub-
tracted from the N value at 1 km above the surface). In both these
tables linear interpolation will suffice for any practical problem. The
variables listed in these tables are for the exponential model of N{h) that
is covered in the following .subsection.
62 TROPOSPHERIC REFRACTION
When the effective earth's radius treatment is used, height is
calculated as a function of distance, for a ray with 60 = 0, with the
equation h = d^/2ka, where d is the distance, k is the effective earth's
radius factor, and a is the true radius of the earth ('^6373 km). The
errors likely to be incurred when using this equation, assuming as a true
atmosphere an exponential N{h) profile as given in the following sub-
section will not exceed 5 percent for heights of 1 km or less.
The preceding background discussion has presented the material neces-
sary for the consideration of the suitability of various models of refrac-
tivity to describe atmospheric refraction of radio waves. As a guide to
what follows, let us ask what a logical sequence of models (or assumptions)
would be to describe the effects of atmospheric refraction.
One such sequence might be:
(1) Assume an invariant model that is near to the actual average
conditions and facilitates the calculation of radio field strengths. This
has been done by the 4/3 earth model.
(2) Assume a variable effective earth's radius factor for the calculation
of radio field strengths in various climatic regions. This approach has
been followed by Norton, Rice, and Vogler [11]. When it has become
apparent that the effective earth's radius approach is inadequate, one
might proceed by :
(3) Correctmg the effective earth's radius model by assuming a more
realistic A^ structure in the region where that model is most in error. This
"modified effective earth's radius" model would then maintain, for some
applications, the advantages of the original model.
(4) Assume an entirely new model of A'^ structure guided by the
average N structure of the atmosphere.
It is assumed that models (3) and (4) would allow for seasonal and
climatic changes of the average refractive index structure of the atmos-
phere.
In the following sections, models (3) and (4) will be developed and
tested by their relative agreement with the ray bendings obtained from
actual long-term average A^^ profiles.
The first model of atmospheric refractivity that will be considered is
based upon the effective earth's radius concept in the first kilometer. In
this atmosphere N is assumed to decay linearly with height from the sur-
face hs to 1 km above the surface hs + 1. This linear decay is given by
N{K) = N, -\- {h - hs) AN,hs < h < hs + I, (3.39)
where
-AA^ = 7.32 exp (0.005577 A^.). (3.40)
MODIFIED EFFECTIVE EARTH'S RADIUS
63
This last relationship comes from the observed relationship between
6 to 8 year averages of daily observations of A'^s and AA^, the difference
between N s and the value of A'' at 1 km above the earth's surface:
-AA^ = A^,, - A^(l km).
It is evident from figure 3.4 that for average conditions, a relationship
exists between AA'' and A^'^. The least squares determination given by
(3.40) was based upon 888 sets of monthly mean values of AA'^ and N s
from 45 United States weather stations representing many diverse cli-
mates. This relationship between AA'^ and N s is expected to represent
the best estimate of a majority of individual profiles and certainly will
closely agree with average conditions for the United States with one
notable exception, southern California in the summer. These data, al-
though shown on figure 3.4, were excluded from the least squares deter-
mination due to their singular large range of AA'' compared to their small
range of A''^ which resulted in a marked "finger" of points rising from the
main body of the data. This obvious departure of data points (24 points
out of a total of 912) plus the well-known unique nature of the southern
California summer climate were taken as sufficient justification for ignor-
ing these points, although, as shall be seen, the ray bendings based upon
-AN
90
80
70
^^_—
60
■ . . ■ -
^
■;, .i
^
^^
7^
'■><'
■ffirf^'
*ir«^
'^■';
\
~^^
rr^r^r,
30
'.v'JJj:,
■^
'i0
M
M
si:^
888 sets of data
20
15
230 240 250 260 270 280 290 300 310 320 330 340 350 360 370 380 390 400
Figure 3.4. Eighl-year average N, versus 6-year average AN at 0300 and 1500 GMT
64 TROPOSPHERIC REFRACTION
the AA^^ obtained from (3.40) are in rather good agreement with the values
calculated from actual A'' profiles, even those observed in southern
California.
Equation 3.40 offers a convenient method of specifying various models
of the refractivity structure of the atmosphere, since it allows an estima-
tion of the value of A^ at 1 km in addition to the two values already
known; i.e., A''^ and A'' = at /i = oo.
It may be further assumed that A'^ decreases exponentially from hs + 1
to a constant value of 105 at 9 km above sea level. In this altitude
range N is defined by:
A^ = A^i exp \-c{h - h, - l)\,hs + I < h < 9 km, (3.41)
where
^ S - h, 105 '
and A^^i is the value of A'' at 1 km above the surface.
Above the altitude of 9 km, where less than 10 percent of the total
bending occurs, a single exponential decrease of A'^ may be assumed. The
coefficients in the exponential expression:
A^ = 105 exp { -0.1424 (h - 9)},h> 9 km, (3.42)
were determined by a least squares analysis of The Rocket Panel data
[12]. This expression is also in agreement with the ARDC Model Atmos-
phere 1956 [13] and Dubin's [14] conclusion that a standard density-
distribution may be used to determine the refractivity distribution at
altitudes in excess of 20,000 ft.
The three-part model of the atmosphere expressed by (3.39)-(3.42) has
the advantage of the effective earth's radius approach, particularly for
such applications as point-to-point radio relaying over distances up to,
say, 100 mi, where the radio energy is generally confined to the first
kilometer, plus being in reasonably good agreement with the average A^
structure of the atmosphere. The reader is cautioned, however, that
application of this model to mode-theory calculations would be mislead-
ing, since the resultant diffraction region fields would be enhanced by the
addition of strong reflections from the r?-gradient discontinuities at /is + 1
and at 9 km. The specific combinations of A^s, hs, and AA^, that define
the CRPL Reference Atmosphere — 1958 are given in table 3.2.
The station elevations corresponding to given combinations of N s and
AA'^ were chosen to correspond with an average decay of A^ with station
elevation. Although the error in neglecting this height dependence has
EXPONENTIAL MODEL 65
Table 3.2. Constants for the CRPL Reference Atmosphere — 1958
N,
/!,
a'
-AN
k
Oe
c
ft
mi
mi
per km
3960. 0000
1.00000
3960. 00
200
10,000
3961.8939
22. 3318
1. 16599
4619. 53
0. 106211
250 _-..
5,000
3960. 9470
29. 5124
1. 23165
4878. 50
0.114559
301
1,000
3960. 1894
39. 2320
1.33327
5280. 00
0. 118710
313.. _.
700
3960. 1324
41.9388
1.36479
5403. 88
0. 121796
350
3960. 0000
51.5530
1. 48905
5896. 66
0. 130579
400-.
3960. 0000
68. 1295
1. 76684
6996. 67
0. 143848
450
3960. 0000
90. 0406
2. 34506
9286. 44
0. 154004
Note: Oe is the effective earth's radius and is equal to a'k, a' = a + hs, where hs is the altitude of the earth's
surface above sea level, a = 3960 miles and c = 1/8— /is In Ni/105.
been estimated to be no more than a few percent, it could be important
in such high precision applications as radar tracking of earth satellites.
It should be remembered in subsequent applications that a unique feature
of these reference atmospheres is the dependence of N s on the altitude of
surface above sea level. This feature was built in so that the reference
atmospheres would be completely specified by the single parameter N s.
3.8. The Exponential Model
The next model of the atmosphere to be considered may be specified
by assuming a single exponential distribution of A^:
where
N ^ Ns exp {-Ce {h - hs)],
Ce = In
Ns
N{1 km)
In
Ns
Ns + AN
(3.43)
(3.44)
and these equations are used to determine A'^ at all heights. This model
of atmospheric refractivity is a close representation of the average re-
fractivity structure within the first 3 km. Further, the single exponential
model has the advantage of being an entire function, and therefore is easily
used in theoretical studies. This model of the atmosphere has been
adopted for use within the National Bureau of Standards with specific
values of the constants in (3.43) and (3.44). These constants are given in
table 3.3 and specify the CRPL Exponential Reference Atmosphere —
1958.
Figure 3.5 compares the A'^ structure of the above two models plus the
4/3 earth model. It can be seen that the assumption agrees with the
reference atmosphere in the first kilometer, which is to be expected since
A'^g = 301 is the value required to yield the 4/3 gradient from figure 3.5.
Figure 3.5 illustrates the essential agreement of the reference atmosphere
66 TROPOSPHERIC REFRACTION
Table 3.3. Table of the constant Ce for the CRPL exponential radio refractivity
atmospheres
N=1S
^exp [— c,(ft
-h,)]
AN
Ns
Ce
(per km)
22. 3318
200.0
0. 118400
29.5124
250.0
. 125625
30. 0000
252.9
. 126255
39. 2320
301.0
. 139632
41.9388
313.0
. 143859
50. 0000
344.5
. 156805
51. 5530
350.0
. 159336
60. 0000
377.2
. 173233
68. 1295
400.0
. 186720
70. 0000
404.9
. 189829
90. 0406
450.0
. 223256
with the Rocket Panel and ARDC data. The exponential reference at-
mosphere is also shown on figure 3.5 for N s = 313, the average value of
the United States. The exponential reference atmosphere appears to be
a reasonable single line representation of A'^ throughout the height interval
shown. The differences between the various models becomes more
apparent by examining their agreement with observed N profiles over the
first 10 km as in figure 3.6.
The reference and exponential reference atmospheres are given for the
N profiles corresponding to near-maximum N s (Lake Charles, La.) and
near minimum-at-sea-level Ns conditions (Caribou, Me.). The two
reference atmospheres were determined solely from the N s values of each
profile. Several observations can be made of these data. First, the 4/3
earth model closely represents the slope of the minimal N s profile over
the first kilometer, but then decreases too rapidly with height. Note,
however, that the 4/3 earth model with its constant decay of 39.2 N units
per kilometer would be a very poor representation of the maximum pro-
file which decreases over 66 N units in the first kilometer. The exponen-
tial reference atmosphere is in good agreement with the initial N distribu-
tion but tends to give values systematically low above approximately
3 km. At first glance, the exponential reference atmosphere does not
appear to be as good a representation of the two observed profiles as the
reference atmosphere, particularly above approximately 5 km. Subse-
quent analysis of the refraction obtained from the two model atmospheres
will show that this systematic disagreement of the exponential reference
atmosphere in the 5- to 20-km interval is a minor defect of the model
compared to its closer agreement with observed N distributions over the
first 1 to 3 km. This is particularly true for the higher values of N s such
as that for Lake Charles.
The above models are more in agreement with long-term mean N pro-
files than is the 4/3 earth model. The application at hand would aid in
deciding which of the reference atmospheres would be most useful. To
EXPONENTIAL MODEL
67
o
Q
<
500
300
200
100
70
50
30
20
10
7
5
3
2
0.7
0.5
0.3
02
0.1
1 '
1
1 I.I 1
RFFFRFMr.F
1 1 1 ! ' ' 1 1 1 r- ■
RFFRflr.TIVITY ATMnc;PHFRF
1
V „ f-, f-,, f
1 ~ " — 1 1 1
\ ICRPL reference
Height Interval
Expression for N |J
^|«/ 1 refractivity
^^v atmosDhere -1958
hjS h< Pj+I
N=N5-{h-hg) AN
hs+l<h< 9km
^^!
' N^^ 301
350
N=N,exp{-(g.',^i'n ^^ )(h-h3-l)}
Nr^
'^\
HUU
h>9km
N = I05 exp {-0.1424 {h-9)}
"T
S>
1
—
—
\
S'
V
1
\
sHi.
et Panel
\
N'nV
V
^
1 'II
6
^N= 313 exp (-0.1439 h)
^\
\
\
\
\
\
\
~
-^
Least Squares fit to Rocket Panel data
^
forh > 9 km- adjusted to pass through
^
'
N= 105 at 9 km
\
1
- —
- N UistriOution tor a
4/3 Earth Amosphere
^\
^
\
k\
rCRPL reference refractivity
Atmosphere -1958
_! 1 1 1 1 1 1
\
K
n
\
^
s.
Ati
Ah
■no.
sph
7 n
ere
wd
el-
95
s
^
V
~'
Vr.
j/j
exf.
iH
1.14
59
h)
r
\V
^s, v
N^vN
\,
CRPL exponential reference "
atmosphere - 1958
O
,
'^
X
^
^
\
\
\
\
%
\
\
4 8 12 16 20 24 28 32 36 40 44 48 52 56 60
ALTITUDE IN KILOMETERS ABOVE SEA LEVEL
Figure 3.5. CRPL reference refractivity atmospheres — 1958.
aid in distinguishing between the various models, the following sections
will be concerned with comparing the ray bending between the models.
A comparison of the ray paths in the reference atmospheres in the 4/3
earth model will illustrate the systematic differences between these
models. Such a comparison is given in figure 3.7 for a distance of 200 mi
and a height of 14,000 ft and in figure 3.8 for a distance of 800 mi and a
height of 240,000 ft. The particular graphical presentation used in
figures 3.7 and 3.8 shows the 4/3 earth rays as straight lines. It is noted
that the 4/3 earth ray at 0o = is in relatively good agreement with the
values from the reference atmosphere for distances out to 200 mi and
heights up to 14,000 ft, but systematically departs from tlu^ n^ference
68
TROPOSPHERIC REFRACTION
m
!
\.
1
1 1
360
340
Reference atmosphere
Exponential reference
atmosphere
\
^
320
300
280
260
240
??0
\
V
\
\
NX
\
^\
\
\ \
\ \\
X/
200
//Vy = 384.1 Lake Charles, La.
July, 03:00 G.M.T
^
\\\
\ "^
\ \
\ \
\\>
1 1
N^^ 302.8 Caribou, Me.
March, 03:00 CM.T.
180
160
140
"1
\
\
A
\
^
\4/J Earth\
120
100
90
^
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\ \
V
2 3 4 5 6 7
HEIGHT IN KILOMETERS
Figure 3.6. Comparison of reference atmospheres with observed N profiles.
EXPONENTIAL MODEL
69
atmosphere for greater distances and heights. For a range of 600 mi,
where the ray reaches heights of about 200,000 ft, the 4/3 earth ray is
some 9000 ft lower than the N s = 400 reference atmosphere and 36,000 ft
lower than the N s = 250 reference atmosphere. This height discrepancy
is due to the 4/3 earth model's unrealistically large A'^ gradient at great
heights with resultant increased bending.
,"^3e^.^
Figure 3.7. Comparison of rays in the CRPL reference refradivity atmospheres — 1968
and the 4/3 earth atmosphere.
t^=S'
3.8. Comparison of rays in the CRPL reference refradivity atmospheres — 1968
and the 4/3 earth atmosphere.
70
TROPOSPHERIC REFRACTION
en
<
<
CC
T at 70 km
Mean profiles: 15.12 mrad
Mean exp. ref.; 15.23 mrad
4/3 earth ; 30.25 mrad
6 8 10 12 14
h-hs IN KILOMETERS
Figure 3.9. Bending versus height.
16
20
Further, the bending in the 4/3 earth atmosphere is compared with
that in the exponential reference atmosphere in figure 3.9. The bending
in an "average" atmosphere is also given. This average atmosphere is
a composite of the 5-year mean profiles for both summer and winter at
the 11 U.S. radiosonde stations enumerated in the following paragraph,
and was used as a readily available measure of average conditions. The
important point made by figure 3.9 is that the 4/3 earth model is sys-
tematically in disagreement with average bending; at low heights it gives
too little bending, while at high altitudes it gives too much bending. The
exponential reference atmosphere does not appear to be systematically
biased, and deviates less than 5 percent from the average atmosphere.
It is significant that the exponential reference and the average atmosphere
are in essential agreement as to the shape of the r-height curve.
It would now be instructive to compare the bendings obtained from
the various models with values obtained from each of the 5-year mean
A^ profiles from different climatic regions. The 5-year mean A'^ profiles
were obtained for both summer and winter for a variety of climates as
represented by the states of Florida, Texas, Maine, Illinois, Nevada,
California, North Dakota, Washington, Nebraska, Wyoming, and by the
District of Columbia.
Comparisons of the bending obtained from the 4/3 earth model and
the bendings obtained from the 5-year mean A^ profiles with the reference
atmospheres are shown in figures 3.10 and 3.11. These figures were
selected to illustrate the range of agreement between the models and the
expected long-term average bendings. Figure 3.10 gives a comparison
EXPONENTIAL MODEL
71
(I)
24
22
20
18
< 16
Q
<
DC
-J 14
^ 12
10
I Washington, D.C.
August, 1500
2 Omaha, Nebr.
August, 1500
3 Santa Maria, Calif
August, 1500
200
300
400
500
N.
Figure 3.10. Comparison of j versus Nb as obtained from CRPL reference atmos-
pheres — 1958 and 5-year mean radiosonde data for ^o = 0, h — he = 3 km.
for a small initial elevation angle, do = 0, and a small height increment,
h — hs = 3 km, and shows that both reference atmospheres tend to set a
lower limit to the bendings. In this case, the exponential reference atmos-
phere appears to be in better agreement with the expected long-term
mean bendings than does the reference atmosphere. The numbered data
points for Washington, D.C, Omaha, Nebr., and Santa Maria, Calif., are
of special interest. Washington and Omaha have the only long-term
72
TROPOSPHERIC REFRACTION
h
1.6
1.5
1.4
(f)
-z.
< 1.3
Q
<
-J 1.2
1.0
0.9
08
0.7
1
/
4/3 Earth Bending = 5.1 nnrad
0o = l5°, h-hs = 70km
/
Exponential ~~~~-.,^_^^^^^ /
Reference ^^~^~W^
Atr
nosphere / !
/^Reference
/
200
300
400
500
Nc
Figure 3.11. Comparison of t versus Ne as obtained from CRPL reference atmos-
pheres — 1958 and 5-year mean radiosonde data for do = 15°, h — ha = 70 km.
mean A^ profiles with initial N gradients (i.e., — 112/km and — 106/km,
respectively) that are significantly greater than would be expected from
the AA^ versus A^^ relationship. Both of these stations have an unusually
large humidity decrease near the ground. The third point, Santa Maria,
Calif., is of interest since it is in relatively good agreement with the refer-
ence atmospheres, even though it represents the southern California
summer climate which was excluded from the original AN versus Ns
EXPONENTIAL MODEL 73
relationship. This agreement is attributed to the fact that the reference
atmosphere is a good representation of the A^ distribution below the
California elevated inversion and to the fact that a majority of the bend-
ing is accomplished below the elevated inversion height of about 500 m.
Further, it can be easily shown that the bending integral is increasingly
insensitive to strong A'' discontinuities as the height increases.
Figure 3.11 shows a similar comparison for a high initial elevation angle,
00 = 15° and a large height increment, h — hs = 70 km. This compari-
son shows that both of the reference atmospheres are in closer agreement
with the long-term mean bendings than are the 4/3 earth bendings. Note
that, whether r is predicted from A^^ or AN, the 4/3 earth model gives but
a single value of bending that is outside the limits of the values of r ob-
tained from the long-term mean profiles.
In considering the comparisons of figures 3.10 and 3.11, one might ask
if they reflected the form of the basic equation for bending; namely, at
low angles is r determined by the A^ gradient throughout the A'" profile,
and at high angles is r essentially a function of the value of A^ at both ends
of the A^ profile (i.e., the limits of integration). Thus one might expect
the deviations to be smaller if the comparisons were made on the basis of
a function of the A^ gradient such as AA^, particularly for small values of
do. Such a comparison is given by figures 3.12 and 3.13 for the same
initial elevation angles and height increment as before. It is seen that
the AA^-specified reference atmospheres improve the agreement for the
low-angle case, but decidedly decrease the agreement for the high-angle
case.
A numerical evaluation of the root mean scjuare (rms) deviation of the
long-term mean bendings from both the reference atmospheres deter-
mined as a function of both AA^ and A''^ was made for a variety of initial
elevation angles for the height increments 3 and 70 km. Root mean
square deviations were not calculated for the 4/3 earth model since it was
felt that this model was obviously in marked disagreement with the long-
term mean bendings under these conditions. Figure 3.14 summarizes
the rms deviations for the h — hs = 3-km case. It is seen that for
do < 10 mrad (about 0.5°), the AA^-specified reference atmospheres have
the smaller rms deviations. Also, the exponential reference atmospheres,
whether specified by AA'^ or A^^, have smaller rms deviations than the
reference atmosphere.
It is seen for the 70-km case, figure 3.15, that the A^s-specified reference
atmospheres have a significantly smaller rms deviation than the AA^-
specified atmospheres for do > 5 mrad. Again it is seen that the exponen-
tial i-eference atmosphere generally has the smaller rms deviation for
values of do less than 10 mrad. Howev(M', the slightly smaller rms devia-
tions associated with the reference atmosphere for ^o > 10 mrad reflect
that model's closer agreement with the actual A structure of the atmos-
phere at high heights.
74
TROPOSPHERIC REFRACTION
C/0
<
<
or
h-
24
22
20
18
14
12
10
1 Washington, D.C.
August, 1500
/
2 Omaha, Nebr.
August, 1500
3 Santa Maria, Calif.
//
August, 1500
.'
2
/
'fc
^, M Mg
Exponeniiui —
Reference
Atmosphere
/
/ /
/ /
•
* / /
,3
•
4
.•/
^Reference
Atmosphere
/
^
/
x^
73 Eai
'th Bending
//
/
20 30 40 50 60 70 80 90
-AN
Figure 3.12. Comparison of t versus AN as obtained from CRPL reference atmos-
pheres — 1958 and 6-year mean radiosonde data for 0o = 0, h. — he = S km.
EXPONENTIAL MODEL
75
C/)
<
<
1.5
1.4
1.2
^ 1.0
0.9
08
0.7
0.6
1 [ ]
/
V
4/3 Earth Bending = 5.1 mrad
E XP'~"^^i^'^i'^l
Reft
Atn
"irence ^*y/^
losphere ^
7^
Reference
<\tmospht
^0=15°
h-hs = '
rakm
3re
• y
•
2
.2
••
»
/ *•
/ •
1
1
Washing
fon,D.C.
1
2
August, IbUU
Onnaha,Nebr.
August, 1500
3
Santa IV
August,
aria,Ca
1500
if.
20 30 40 50 60 70 80 90
-AN
Figure 3.13. Comparison of t versus AN as obtained from CRPL reference atmos-
pheres — 1958 and 6-year mean radiosonde data for do = 15° h — hs = 70 km.
76
TROPOSPHERIC REFRACTION
q: I
0.1
Mean bending is that for
reference atnnosphere Ns=330
•Reference Atmosphere
■Exponential Reference Atnnosphere
0.01
0.1
I 10
IN MILLIRADIANS
100
1000
Figure 3.14. Covi-parison of percent RMS deviations of 5-year mean profile bendings
about CRPL reference atmospheres using two parameters, Ng and AN for
h — hs = S km.
100
Eh
Mean bending is that for
reference atnnosphere Ns = 330
•Reference Atmosphere
-Exponential Reference Atmosphere
1.0 10
IN MILLIRADIANS
1000
Figure 3.15. Comparison of percent RMS deviations of 5-year mean profile bendings
about CRPL reference atmospheres using two parameters, Ns and AN for
h - hs = 70 km.
DEPARTURES-FROM-NORMAL METHOD 77
3.9. The Initial Gradient Correction Method
The importance of the initial gradient in radio propagation, where the
initial elevation angle of a ray path is near zero, has long been recognized.
For example if dn/dh = — 1/a (the reciprocal of the earth's radius), then
the equation for r is indeterminate, an expression of the fact that the
ray path remains at a constant height above the earth's surface. This is
called ducting, or trapping of the radio ray. The effect of anomalous
initial A^-gradients on ray propagation at elevation angles near zero, and
for gradients less than ducting ( | dN/dh \ < 157 N units/km, or dN/dh >
— 157 A^ units/km) may also be quite large. A method has been devel-
oped for correcting the predicted refraction (from the exponential refer-
ence atmosphere) to account for anomalous initial A'^ gradients, assuming
that the actual value of the initial gradient is known [2].
The result is
Th = Th (Ns, So) + [riooiN*, do) - T100{N„ do)], (3.45)
where th (N s,do) = r at height h, for the exponential reference atmosphere
corresponding to A''^, and N s* is the A''s for the exponential reference
atmosphere that has the same initial gradient as the observed intiial
gradient; t^qq is r at a height of 100m.
This procedure has the effect of correcting the predicted bending by
assuming that the observed initial gradient exists throughout a surface
layer 100 m thick, calculating the bending at the top of the 100-m-thick
layer, then assuming that the atmosphere behaves according to the ex-
ponential reference profile corresponding to the observed value of A^^ for
all heights above 100 m. This approach has proved quite successful in
predicting T for initial elevation angles under 10 mrad, and will, of course,
predict trapping when it occurs.
3.10. The Departures-From-Normal Method
A method of calculating bending by the use of the exponential model
of N{h) together with an observed N{h) profile can sometimes be advan-
tageously employed [15]. This method is primarily intended to point out
the difference between actual ray bending and the average bending that is
predicted by the exponential N{h) profile and is a powerful method of
identifying air mass refraction effects.
The exponential model described in section 3.8 can be expected to
represent average refractivity profile characteristics at any given location,
but it cannot be expected to depict accurately any single refractivity pro-
file selected at random, even though it may occasionally do so. In order
78 TROPOSPHERIC REFRACTION
to study the differences between individual observed N(h) profiles and the
mean profiles predicted by the exponential model, a variable called the
A unit has been developed; it is defined simply as the sum of the observed
A'' at any height, h, and the refractivity drop from the surface to the
height, h, which is predicted by the exponential profile for a given value
of A^..
Thus
A{Ns,h) = N(h) + Ns (l-exp{ -c,/i}). (3.46)
Thus (3.46) adds to N{h) the average decrease of N with height, so that
if a particular profile should happen, by coincidence, to be the same as
the corresponding exponential profile, the value of A {N s,h) for this pro-
file would be equal to N s for all heights. The above analysis shows that
the difference between A{Ns,h) from N s, 8A{Ns,h), is a measure of the
departure of N(h) from the normal, exponential profile:
5.4(A^,,^) = AiNs,h) - Ns = A^(/i)— iV. exp{ -c./i}. (3.47)
It seems logical that the application of the A unit to bending would indi-
cate the departures of bending from normal, in some way, just as it indi-
cates departures of refractivity, A'^, from normal. This is indeed the case,
as can be seen in figure 3.16, where for an A'^s = 313.0 exponential atmos-
phere, .4(313.0, h) is plotted on one set of graphs for various typical air
masses, and the corresponding bending departures from normal are shown
in the second set of graphs corresponding to the same air masses. Ob-
viously, the bending departures between layers are highly analogous to
the A unit variation. It can be seen from figure 3.16 that the similarity
exists, although it is less, for higher initial elevation angles. The simi-
larity also decreases with increasing height, owing to the fact that the
bending departures from normal are an integrated effect, and at low initial
elevation angles are more sensitive to A^ variations at the lower heights.
This causes an apparent damping of the bending departures from normal
at greater heights. However, the A-unit variation is not similarly in-
fluenced; hence, a loss of similarity arises at large heights above the earth's
surface.
If (3.46) is differentiated and substituted into (3.2), the following equa-
tion results:
k = Uk -T Uk+1
(rad) [rad) (.rad) (rad)
^A{N.
'k+\
X 10-', (3.48)
1
DEPARTURES-FROM-NORMAL METHOD
79
s
WINTER
CP MP MT cT
WINTER WINTER SUMMER SUMMER
\ ;; :
320 340
1
1
/
"1
-
-)
1
-
-<»
_
r>A
3.h),
N U
I 1
1 1
-
I
\
1
1 -
} '
1
-2 0-2 2-4 -2
T^l^-Tdnrad)
Figure 3.16. A-unit profiles for typical air ynasses and refraction deviation from normal.
where
A.4(A^,) = ANih) + A[iV,|l-exp(-c,/i)!]
= AN{h) + N sCeexp{-Ceh)Ah,
and TNs{h) is the value of r tabulated for various atmospheres in tables
9.10 to 9.17, dk and dk+i are in milliradians and must be from the N s ex-
ponential atmosphere used. The value of AA{N s) is obtained from sub-
traction of the A value at layer level, k, from the value of A at layer, A' + l.
The A value may be obtained by adding any given N{h) value, obtained
from RAOB or other similar data, to a value of A^s[l ~exp{ —ch]] for the
same height which may be obtained from figure 3.17. Since TNs{h) has
been calculated only for a few of the exponential atmospheres, these being
the A^, - 200.0, 252.9, 289.0, 313.0, 344.5, 377.2, 404.9, and 450.0 atmos-
pheres, one of these atmospheres must be used in the calculation of bend-
ing by the departures method. The selection of the particular atmos-
phere to be used is based on the value of the gradient of N, dN/dh, be-
tween the surface of the earth and the first layer considered. In table 3.2
are shown the ranges of the gradient for the choice of a particular ex-
ponential atmosphere.
80
TROPOSPHERIC REFRACTION
50
300
350
100 150 200 250
Ns[l-exp(ch)] IN N UNITS
Figure 3.17. Graphic representation of Ns[l — exp(— ch)] in N iinits versus height.
3.11. A Graphical Method
Weisbrod and Anderson [16] present a handy graphical method for
computing refraction in the troposphere. Rewriting and enlarging (3.11),
one obtains
N,- iV,+i
T(mrad) = 2^ tttttt: z — n: — o — \
k^o 500 (tan dk + tan dk+i)
(3.49)
where r will be the total bending through n layers. Terms for the de-
nominator can be determined from figure 3.18. Equation 3.49 is essen-
tially Schulkin's result with only the approximation, tan dk ~ Ok, for
small angles, omitted.
The procedure for using figure 3.18 follows. Enter on the left margin
at the appropriate Ns — N{h). Proceed horizontally to the proper
height, h, interpolating between curves if necessary. Use the solid height
curves when N s — N{h) is positive and the dashed curves when N s —
N{h) is negative. Then proceed vertically to the assumed ^o and read
500 tan 6 along the right margin.
GRAPHICAL METHOD
81
1000
J 1000
Figure 3.18. Graphic representation of Snelis Law for finding 500 tan d.
82 TROPOSPHERIC REFRACTION
3.12. Derivations
The approximate relation between ^i and 62 is derived here. This rela-
tion holds for small increments of height and small ^'s. The relationship
was used in making all sample computations in preceding sections.
Since for small ^'s
cos ^1^1- ~ and cos 02 = 1 - - , (3.50)
and knowing
r, = ri + ^h (fig. 3.19)
then substituting in (3.1) yields
n,{n + A/i)( 1 - f ) = n:n( 1 - f ) , (3.51)
or
i2
'2 . dl
UiVi + UiAh — riiri -- — n^Ah -- ^ UiTi — UiVi — . (3.52)
Dividing by Vi
. UiAh niQ\ Ah dl ^ e\ ,_ __.
^2 + -^ - -^ - n^ - 2 ^ ni - n. -- . (3.53)
Since the term — n2 (A/i/rO {61/2) is small with respect to the other
terms of (3.53) it may be neglected, and thus:
n2Ah b\ ^ d\
or
n2 + -^— - n2 77 = ni - ni -^ (3.54)
T\ Z Zi
^2 ~ ^1 ^2A/l . , , .
-n2 7,- = -ni — + (ni - n2). (3.55)
If one now divides both sides of (3.55) by ni and assumes
(ni — ri2)/r?2 = Wi — 712 and nxln-i = 1, (3.55) may be arranged to yield
72 ^ x/^i + -— - 2(ni - 71,). (3.56)
DERIVATIONS 83
Writing (3.56) in terms of A^ units,
02(mrad) ^ ^0i + — X lO' - 2iNr - N2) (3.57)
if 01 is in milliradians.
Generalizing (3.57) for the ^'th and the (A: + l)st layers,
0,+i(mrad) ^ xJeUmrsid) + ^"^ ^ X lO' - 2{N, - N,+r). (3.58)
Also from the geometry shown in figure 3.19, a useful relationship for
Ti,2 can be obtained. Tangent lines drawn at A and B will be respec-
tively perpendicular to ri and r-z, since ri and r2 describe spheres of refrac-
tive indices Ui and n2 concentric with 0. Therefore,
angle AEC ^ angle AOB ^ <}>
also, in triangle A EC
angle ACE = 180° - angle CAE - angle A EC
= 180° - di - <t>. (3.59)
But from triangle DCB
angle ACE = angle DCB = 180° - ti,2 - ^2. (3.60)
Thus
180° - ri.2 - 02 = 180° - di - <t>,
or
T1.2 - (^ + (01 - 02). (3.61)
Now since <f> in radians = d/a, where d is distance along the earth's
surface :
T1.2 = - + (01 - 02), (3.62)
a
or the bending of a ray between any two layers is given in terms of the
distance, d, along the earth's surface from the transmitter (or receiver),
the earth's radius, a, and the elevation angles 0i and 02 (in radians) at the
beginning and end of the layer.
84
TROPOSPHERIC REFRACTION
Figure 3.19. Bending geometry on a spherical earth with concentric layers.
If one considers figure 3.20, Snell's law in polar coordinates (3.1) and
the refraction formula (3.2) may be obtained from the more familiar
form of Snell's law.
Assume that the earth is spherical and that the atmosphere is arranged
in spherical layers. In figure 3.20 let C be the center of the earth, the
observer and COZ the direction of his zenith. Let n and n+dn be the
indices of refraction in two adjacent thin layers M and M' . Let LP be
the section of a ray in M' which finally reaches the observer at 0. At
P it is refracted along PQ. Similarly, it is refracted at the surfaces be-
tween successive layers and the final infinitesimal element of its path is
TO.
Draw the radii CP and CQ. Let angle PQF = 6, angle LPS = 6 -\- dd,
and angle QPF = \p. Then, since the radius CP is perpendicular at P to
DERIVATIONS
85
-EARTH'S
SURFACE
Figure 3.20. Geometry for the derivation of Snell's Law in spherical coordinates.
the bounding surface between layers M and M', by Snell's law we have
(n+rfn) sin [90° - id+dd)] = n sin i/^. (3.63)
Now from the triangle CQP-, in which CQ = r and CP = r -\- dr, and
2The assumption involved in this triangle is that the path of the ray in M' is a
straight line, which, of course, can only be true in an isotropic medium. Hence, it
can only be true for an infinitesimal layer in the troposphere. Thus only a differential
form of Snell's law, (3.65), in polar coordinates, can be obtained by the use of the
geometry of figure 3.20; not the finite form, (3.68), which has the same appearance.
86 TROPOSPHERIC REFRACTION
angle CQP = 90° + 6, we have, from the law of sines,
r sin (90° + 6) = r sin (90° - 0) = (r + dr) sin xj^. (3.64)
Eliminating sin i/^ from (3.63) and (3.64) we then have
(n + dn)(r + dr) sin [90° - (^ + dd)] = nr sin (90° - 6),
or
(n + dn)ir + rfr) cos (0 + dd) = nr cos 6. (3.65)
Multiplication of the (n + (in) and (r + dr) terms, ignoring differential
products, yields
{nr + ndr + rdn) cos (0 + dd) = nr cos 0,
or
{nr + n(/r + rdn) [cos ^ cos ((/0) — sin 6 sin (c?0)]
= nr cos 0. (3.66)
Since cos {dd) = 1, and sin {dd) ^ c?^, another multiplication, again ignor-
ing products of differentials, yields:
ndr cos d + rdn cos d — nr sin d dd = 0,
or, dividing all terms by nrcos 0,
- + — -ta^nddd = 0. (3.67)
r n
Now if (3.67) is integrated between any two thin layers of refractive
indices ni and n2, whose radial distances from the earth's center are ri and
r2, and the initial elevation angles of a radio ray entering the layers are di
and 02:
dr . dn / . «7^ . ^2 , , n2 , , cos ^2 ^
[- / — — / tan ddd = (n j- ^n — + ^n =
, r Jn, n Je, ri ni cos ^i
or, taking antilogs of both sides,
r2n2 cos 02
rini cos di
= 1,
REFERENCES 87
whence
riiri co.s di = ti2r2 cos do, (3.68)
which is Snell's law for polar coordinates, (3.1).
In figure 3.20, it can be seen that
PF dr
'^"'=QF = 7d-^- (3.69)
where </> is the angle at the earth's center between r and COZ. Substitut-
ing (3.69) in (3.67)
dn
tan dd(f) -\ tan Odd = 0,
n
or
dn
id4> - dd) tan = - — . (3.70)
Since, by considering (3.61) for infinitesimal angles,
dr = d<t) - de,
or, in (3.70)
or
dr tan 6 = — —
n
dti
dr = -cote — . (3.71)
n
Integration of (3.71) yields (3.2).
3.13. References
[1] Smart, W. M. (1931), Book, Spherical Astronomy, Ch. 3 (Cambridge Univ.
Press, London, England).
[2] Bean, B. R., and G. D. Thayer (May 1959), On models of the atmospheric re-
fractive index, Proc. IRE 47. No. 5, 740-755.
[3] Booker, H. G., and W. Walkinshaw (1947), The mode theory of tropospheric
refraction and its relation to wave guides and diffraction, Book, Meteorological
Factors in Radio-Wave Propagation, pp. 80-127 (The Physical Society,
London, England).
[4] Freehafer, John E. (1951), Tropospheric refraction. Book, Propagation of Short
Radio Waves, pp. 9-22 (McGraw-Hill Book Co., Inc. New York, N.Y.).
[5] Bean, B. R., and B. A. Cahoon (Nov. 1957), The use of surface weather observa-
tions to predict the total atmospheric bending of radio waves at small elevation
angles, Proc. IRE, 45, 1545-1546.
88 TROPOSPHERIC REFRACTION
[6] Schulkin, M. (May 1952), Average radio-ray refraction in the lower atmosphere,
Proc. IRE 40, 554-561.
[7] Anderson, L. J. (Apr. 1958), Tropospheric bending of radio waves, Trans. Am.
Geophys. Union 39, 208-212.
[8] Schelleng, J. C, C. R. Burrows, and E. B. Ferrell (Mar. 1933), Ultra-short-wave
propagation, Proc. IRE 21, 427-463.
[9] Stickland, A. C. (1947), Refraction in the lower atmosphere and its application to
the propagation of radio waves. Book, Meteorological Factors in Radio Wave
Propagation, pp. 253-267 (The Physical Society, London, England).
[10] Humphreys, W. J. (1940), Book, Physics of the Air, p. 82 (McGraw-Hill Book
Co., Inc., New York, N.Y.).
[11] Norton, K. A., P. L. Rice, and L. E. Vogler (Oct. 1955), Use of angular distance
in estimating transmission loss and fading range for propagation through a
turbulent atmosphere over irregular terrain, Proc. IRE 43, 1488-1526.
[12] The Rocket Panel (1952), Pressures, densities, and temperatures in the upper
atmosphere, Phys. Rev. 88, 1027-1032.
[13] Handbook of Geophysics for Air Force Designers (1957), Geophysics Research
Directorate (Air Force Cambridge Research Center, ARDC, USAF).
[14] Dubin, M. (Sept. 1954), Index of refraction above 20,000 feet, J. Geophys. Res.
59. 339-344.
[15] Bean, B. R., and E. J. Dutton (May-June 1960), On the calculation of departures
of radio wave bending from normal, J. Res. NBS 64D (Radio Prop.), No. 3,
259-263.
[16] Weisbrod, S., and L. J. Anderson (Oct. 1959), Simple methods for computing
tropospheric and ionospheric refractive effects on radio waves, Proc. IRE 47,
1770-1777.
Chapter 4. N Climatology
4.1. Introduction
The contents of this chapter include a study of the surface variation of
the radio refractive index on a worldwide scale in terms of a reduced-to-
sea-level form of the index that gives a significantly more accurate descrip-
tion of refractive index variations than the nonreduced form.
The mean vertical structure of the refractive index parameter in the
troposphere over central North America is presented, again in terms of
a reduced-to-sea-level form of A^.
A climatological treatment of the phenomenon of the atmospheric duct,
or waveguide, and associated fading regions is also presented. The
chapter is concluded with a discussion of refraction of radio waves in
various air masses. It is demonstrated that refraction differences within
air masses arise from departures of refractive index structure from normal.
4.2. Radio-Refractive-Index Climate Near the Ground
4.2.1. Introduction
The radio refractive index of air, w, is a function of atmospheric pres-
sure, temperature, and humidity, thus combining in one parameter three
of the normal meteorological elements used to specify climate. In the
following sections we will examine the variability of n during different
seasons of the year and in differing climatic regions. The systematic
dependence of n upon station elevation will make it necessary to consider
a method of expressing n in terms of an equivalent sea-level value in
order to see more clearly the actual climatic differences of the various
parts of the world. After a consideration of the n climate of the world,
the application of this information to such practical problems as the
prediction of radio field strength and the refraction of radio waves will be
discussed.
4.2.2. Presentation of Basic Data
Near the surface of the earth, for VHF and UHF frequencies, n is a
number of the order of 1.0003. Since, for air, n never exceeds unity by
89
90
N CLIMATOLOGY
more than a few parts in 10^. it is convenient to consider the chmatic
variation of n in terms of
N = {n - 1)10«,
(4.1)
as defined in chapter I. The notation N s is used to indicate that (4.1)
has been evaluated from standard surface weather observation.
To obtain long-term average values of A^, one should properly average
individual observations over many years. This is difficult to do since,
in general, only summaries of weather observations are readily available.
However, long-term average values of temperature, pressure, and
humidity are available and may be converted into an "average" value
of A''. This "average" N differs from the true average since the inter-
correlation of pressure, temperature, and humidity is neglected. This
difference was examined by an analysis of 2 years of weather records of
the months of February and August at an arctic location (Fairbanks,
Alaska), a temperate zone location (Washington, D.C.), and a tropical
location (Swan Island, W.I.). These data, given in table 4.1, indicate
that the difference between the two methods was never more than 1.5 A''
units and that the average difference was less than 1 A'' unit, which is small
compared to commonly observed seasonal and geographic variations of
20 to 100 A^ units.
Table 4.L Two-year average value of Ns versus the value of Ns calculated from average
temperature, pressure, and humidity
Fairbanks;
February.
August
Washington:
February.
August
Swan Island:
February.
August
N,
(P,
T,
RH)
N,-N, (P, T, EH)
314.0
320.5
313.0
320.0
1.0
0.5
305.5
356.0
340.5
354.5
1.0
1.5
362.0
387.5
362.5
388.0
0.5
0.5
Average.
0.83
On this basis it was decided to use the long-term means given in the
United Nations' monthly publication, Climatic Data for the World. This
publication is particularly advantageous for our present study since it re-
ports the fictitious value of the relative humidity needed to obtain the
actual average vapor pressure from the saturated vapor pressure of the
reported mean temperature [1].^
^Figures in brackets indicate the literature references on p. 170.
PRESENTATION OF BASIC DATA 91
Data from 306 weather stations were obtained in order to give reason-
able geographical coverage. In general, 5 years of records were obtained
for each station from the period 1949 to 1958, preference being given to
the years 1954 through 1958. A noticeable exception, however, was
Russia, for which only 1 year of data (IGY) is reported in Climatic Data
for the World; thus all charts are drawn with dashed contours for Russia.
There are vast expanses of ocean for which there are no meteorological
observing stations. Climatic atlases were utilized in order to present
estimates of world climate in these locales. A reasonable coverage of
the sparse data areas of the world was made by estimating temperature
from sea surface isotherms [2] and humidity from charts of seasonal
average depression of the wet bulb temperature [3]. Pressure was esti-
mated for these locations from average winter and summer pressure
charts.
When these data were converted to A^ [4] and charts prepared, a pro-
nounced altitude dependence could be seen, as in figure 4.1. Figure 4.1
and the following charts of N variations across the United States are from
an extensive N climatology now being prepared at the Central Radio
Propragation Laboratory. Although the present study is primarily
aimed toward worldwide variations, it is felt that the U.S. data better
illustrate the height dependence of Ng and the subsequent reduction
process employed. It is noted that the coastal areas display high values of
N s, while the inland areas have lower values. There are low values of
N s corresponding to the Appalachian and Adirondack Mountains and a
decrease with increasing elevation of the Great Plains until the lowest
values are observed in the Rocky Mountain region and the high plateau
area of Nevada. A corresponding gradient is observed from the west
coast eastward. Crosshatching encloses areas where the terrain changes
so rapidly that it was felt the data were inadequate to obtain realistic
contours of N s.
The altitude dependence of N can be studied in terms of the "dry" and
"wet" components of N. These components are those of the two-term
expression in (1.17). The dry term, D,
i) = ^^ , (4.2)
is proportional to air density and normally constitutes at least 60 percent
of A.
The average variation of density with altitude in the atmosphere may
be expressed in the first approximation as
p = pocxp {-z/H\ (4.3)
92
N CLIMATOLOGY
Figure 4.L Mean Ns, August 0200 local time.
where z is the akitude, po the average sea level density of moist air and
H the average scale height between zero and z. It is useful to introduce
the concept of an effective scale height, H*, for the average variation of
refractive index in the atmosphere. Many studies have shown that the
average refractive index variation with height is quite well represented,
to a first approximation, by a formula similar to (4.3) [5, 6, 7]. It is
possible to calculate a theoretical value of this effective scale height using
a distribution of water vapor. This is, however, quite a complex pro-
cedure. Furthermore, the value obtained depends upon the model of the
water vapor distribution, and no definite conclusion can be justified
considering the extreme variation of water vapor concentration with
season, geographic location, and height above the earth's surface. A
convenient and simple alternative is to adopt a value for H* from the
average (n — 1) variation with height in the free atmosphere. Several
such values of H* were determined by reference to the NACA standard
atmosphere [8] and recent climatological studies of atmospheric refractive
index structure [9].
It is seen from table 1.8, chapter 1, that H* varies from 6.56 to 7.63
km in the NACA standard atmosphere, depending on the value of rela-
tive humidity assumed. The value of H* = 6.95 km obtained from
chmatological studies of (n - 1) variations over the first kilometer above
PRESENTATION OF BASIC DATA 93
the earth's surface from nearly 2 million radiosonde observations from
many diverse climates.
At the time of analysis of the map series presented here, a value of
effective scale height, H* = 9.46 km, was in use. This form was
Do = D, exp j^l
(4.4)
in which the value H* = 9.46 km was determined from the NACA dry
standard atmosphere. A Do chart is shown on figure 4.2, presenting a
gradient that is remarkably free of detail as compared to the N ^ chart of
figure 4.1, and is easily drawn for all areas of the country.
An investigation of the elevation dependence of the surface wet term
W. = ^^A^^ (4.5)
revealed low correlations of log W s and height, indicating that W s is not
a marked exponential function of elevation. Contours of Ws for all
sections of the country are shown on figure 4.3.
The maps that follow are completed in terms of a single reduced form
A^o ^ {Ds + Ws) exp j^^l =^ Ns exp j^[, (4.6)
where N s is reduced by the dry term effective scale height, H* = 9.46 km.
Figure 4.4 gives the A^o contours for the same time as the previous maps
of Do and W s- The A^o maps are no more difficult to prepare than the W s
maps and have effectively removed the station height dependence of N s.
One might wonder at the advisability of arbitrarily reducing the wet term
by the dry term correction. For the coastal areas of the country, where
the exponential height correction factor is nearly unity, this amounts
simply to adding the Do and W s maps for the mountain areas. Where the
height correction factor is large, the W s values are small with the result
that the gradient of the N isopleths obtained from the Do and W s maps is
essentially maintained on the Ao maps. As an example, for the series of
maps under discussion, the {Do -\- W s) difference between Reno, Nev.
(1,340 m elevation), and Oakland, Calif. (5.5 m elevation), is 21 N units,
while the Ao difference is 19 A^ units.
The effects of this correction on the worldwide values can be seen from
figure 4.5, where As is plotted versus station elevation in kilometers. A
sample line illustrates the decay of As with height for Ao = 348. The
value of As for any other value of A^o would be obtained from a line parallel
\o the Ao = 348 line but having a zero intercept equal to the new value of
94
A^ CLIMATOLOGY
Figure 4.2. Mean Do, Aug^^st 0200 local time.
Figure 4.3. Mean We, August 0200 local time.
PRESENTATION OF BASIC DATA
95
Figure 4.4. Mean No, August 0200 local time.
No. The advantage of adopting No is illustrated by the reduction in
range from 190 A^ units for iV^ to 115 A^ units for No, thus diminishing
the number of contours of the resulting maps.
It would appear that by removing the influence of station elevation it
would be more efficient to estimate A''^ from A^^o charts rather than from
charts of A''^. As a test of this hypothesis, N s and A^'o contour charts were
prepared for both summer and winter from only 42 of the 62 U.S. Weather
Bureau stations for which 8-year means of N s are available. The remain-
ing 20 stations, distributed at random about the country, were used as a
test sample by estimating their 8-year mean value of N s from the A'^o and
N s contours. Summertime examples of these charts are given by figures
4.6 and 4.7. Note that due to the reduced range of N, the A^'o charts are
drawn every 5 A'^ units as compared to the 10-A^-unit contours of the N s
charts. The individual deviations of the values obtained from the con-
tour maps with the actual 8-year means are listed in table 4.2. By com-
paring the root mean square (rms) deviations of 10.7 A'^ units in winter
and 13.0 N units in summer obtained by estimating N s from the N s con-
tours with the 2.7-A^-unit rms of estimating N s from A'^o contours, one
concludes that it is at least 4 times more accurate to estimate N s from A''o
contours than from those of A^^. An inspection of the individual devia-
tions in table 4.2 indicates that the A^o contour method is particularly
efficient at elevations in excess of 1,200 m or where the terrain is changing
96
A^ CLIMATOLOGY
0.5 1.0 1.5 2.0 2.5 3.0
STATION ELEVATION IN KILOMETERS
Figure 4.5. Worldwide values of Ns versus height for August.
PRESENTATION OF BASIC DATA
97
Figure 4.6. Test chart of mean Ns August 0200 local time.
Figure 4.7. Test chart of mean No August 0200 local time.
98
N CLIMATOLOGY
rapidly with respect to horizontal distance. As a further practical con-
sequence, one notes the remarkable similarity between the A^'o contours
of figures 4.4 and 4.7, even though the latter contours were derived from
only two-thirds of the original data. This indicates that any desired level
of accuracy may be maintained with fewer stations (and less expense) by
the use of No.
Table 4.2. Deviations of estimated 8-year means of Ns {calculated from contour charts)
from actual 8-year means selected at random from 20 Weather Bureau stations
Test station
Sacramento, Calif..
Portland, Oreg
San Diego, Calif...
Mobile, Ala
Fresno, Calif
Boston, Mass
Grand Rapids, Mich.
Columbia, Mo
Minneapolis, Minn...
Cincinnati, Ohio
Des Moines, Iowa
Pendleton, Oreg
Billings, Mont
Burns, Oreg
Salt Lake City, Utah.
Reno, Nev
Pocatello, Idaho
Denver, Colo
Colorado Springs, Colo.
Flagstaff, Ariz
Root mean square deviation.
Height
meters
7
8
11
66
86
210
239
255
271
294
455
1,088
1,262
1,288
1.340
1,355
1,625
1.882
2,131
February 1400
Actual
8-yr
mean
value
of JV,
N units
315.6
316.2
314.2
326.6
310.6
308.6
304.4
300.8
301.1
302.5
300.9
295.9
269.3
268.1
266.3
259.6
264.7
244.9
237.1
237.8
Deviation*
map
N tinits
7.6
4.2
-5.8
6.6
9.6
-6.4
0.4
-.2
.1
-.5
3.9
1.9
-2.3
-23.9
1.3
-20.4
-2.3
-8.1
-15.9
-26.2
10.7
No
map
A'^ units
0.8
-.5
-2.4
4.8
3.4
-0.5
-5.9
2.4
0.6
.7
2.3
0.4
.1
-3.2
-0.8
-6.8
0.4
.7
-.6
2.3
2.7
August 0200
Actual
8-yr
mean
value
of iV,
A'' units
329.6
337.7
348.1
376.0
326.2
347.5
340.5
348.7
338.5
344.1
343.1
300.9
285.6
271.3
279.5
277.6
269.7
276.6
272.4
261.4
Deviation
Ns
map
A'' units
1.0
19.7
16.1
6.0
5.2
-7.5
-1.5
-2.3
-0.5
-2.9
-1.9
2.9
5.6
-15.7
8.5
-29.4
-3.3
-1.4
-6.6
-36.6
13.0
ATo
map
A^ units
1.8
6.0
3.5
0.6
4.2
-0.4
.1
-2.5
2.7
-2.8
-0.1
-3.1
1.2
-4.4
4.4
1.6
0.0
.3
1.1
-2.2
2.7
♦Deviation equals the actual long-term mean minus the value obtained from map contours.
4.2.3. Worldwide Values of N„
Mean values of No were calculated at each of the 306 selected stations
and charts were prepared for each month of the year. The charts for
February and August are given on figures 4.8 and 4.9. It is seen that
the values of No for sea-level stations vary from 390 in the maritime
tropical areas to 290 in the deserts and high plateaus. The interiors of
continents and mountain chains in middle latitudes are reflected by low
values as compared to those of coastal areas. Further, such pronounced
climatic details as the Indian monsoon and the effects of coastal mountain
ranges blocking prevailing winds and producing rain shadows are indi-
cated by these No contours. Regional climatological data of No for the
United States are given in chapter 9.
WORLDWIDE VALUES OF A^o
99
Figure 4.8. Worldwide values of mean No /or February.
Figure 4.9. Worldwide values of mean 'No for August.
100
A^ CLIMATOLOGY
Figure 4.10. Annual range of monthly mean Ns.
Figure 4.n. Minimum monthly mean value o/No.
WORLDWIDE VALUES OF A^o
101
Figure 4.12. Year-to-year range of monthly mean Ng/or February.
P'iGURE 4.13. Year-to-year range of monthly mean Na/or August.
102 N CLIMATOLOGY
The annual variation of N s is indicated on figure 4.10 by contours of
the difference between the maximum and minimum monthly means ob-
served throughout the year. It is quite remarkable how clearly climatic
differences are evidenced by the yearly range of N s. The prevailing
transport of moist maritime air inland over the west coasts of North
America and Europe is indicated by relatively small annual ranges (20 to
30 A'^ units), while, for example, the east coast of the United States with a
range of 40 to 50 A^ units or more reflects the invasion of that area from
time to time by such diverse air masses as arctic continental and tropical
maritime. The largest annual ranges of N ^ (90 N units) are observed in
the Sudan of Africa and in connection with the Indian monsoon.
An additional A'^o map (fig. 4.11) was prepared from the minimum
monthly mean value of A^^ observed throughout the year to supplement
the range map in order that an estimation might be made of both the
minimum and maximum monthly mean N s expected durmg the year.
A measure of the variability of the February and August mean values
of A'"^ is given by monthly range maps (fig. 4.12 and 4.13) determined
from monthly averages from 5 years of data. Ranges are given by the
maximum difference of the five individual monthly mean values. In con-
touring the two variability maps only those terrestrial regions having
reasonable data coverage are included. Dashed contours are shown for
areas of sparse or unreliable data. The general picture of the worldwide
distribution of A''^ variability is that of a number of continentally located
cells of moderate range accompanied by somewhat random small-scale
variation over ocean areas. Regions of large range, from 40 to as much
as 70 A^ units, are present, however, in Australia and on islands of the
adjoining oceans, on the African equatorial plateau near the Cameroons,
and in the Great Basin of the southwestern United States. Common to
all these areas of large year-to-year variability, at least during the summer
season, are high mean temperatures ranging from about 25 to 30 °C, the
variability being due to relatively small variations of humidity. It is felt
that when a more dense network of stations is available for a longer period
of record, say 10 years, areas of high monthly variability are likely to be
more extensive in tropical and desert areas than indicated on our present
maps.
4.2.4. Climatic Classification by N^
The annual cycle of A^« at each station was examined for the purpose
of deriving similarities of climatic pattern. As one form of climatic
classification, the annual mean value of A'^s at each station was plotted
versus the annual range at the station. When this was done, several
distinct groupings of data seemed evident. These groupings, described
CLIMATIC CLASSIFICATION BY No
103
in table 4.3, are intended to give a general idea of the geographic and
climatic character of the majority of the stations found within given
values of range and yearly mean of N s.
Table 4.3. Characteristics of climatic types
Type
Location
Annual
mean N,
Annual
range of A'',
Characteristics
I. Midlatitude-
coastal.
Near the sea or in lowlands on
lakes and rivers, in latitude
belts between 20° and 50°.
N units
300 to 350
N units
30 to 60
Generally subtropical
with marine or modi-
fied marine climate.
n. Subtropical-
Savanna.
Lowland stations between 30°N
and 25°S, rarely far from the
ocean.
350 to 400
30 to 60
Definite rainy and dry
seasons, typical of
Savanna climate.
III. Monsoon-
Sudan.
Monsoon — generally between
20° and 40°N, Sudan— across
central Africa from 10° to 20°N _
280 to 400
60 to 100
Seasonal extremes of
rainfall and tem-
perature.
IV. Semiarid-
mountain.
In desert and high steppe regions
as well as mountainous re-
gions above 3,000 ft.
240 to 300
to 60
Year-round dry climate
V. Continental-
Polar.
In middle latitudes and polar
regions. (Mediterranean cli-
mates are included because of
the low range resulting from
characteristic dry summers.)
300 to 340
to 30
Moderate or low annual
mean temperatures.
VI. Isothermal-
equatorial.
Tropical stations at low eleva-
tions between 20°N and 20°S,
almost exclusively along sea-
coasts or on islands.
340 to 400
to30
Monotonous rainy
climates.
For a given classification of refractive-index climate, diverse meteor-
ological climates and geographical regions may be represented. Note,
for example, that type V of table 4.3 includes stations from Mediterranean
and marine as well as polar climates. Mediterranean stations in this
category fail to attain a high range because of the characteristic dryness
of the subtropical high-pressure pattern that is generally found in this
area during the summer months. Polar and marine climates in this
group maintain a low range due to suppressed humidity effects as a result
of low to moderate year-around average temperatures.
Annual trends of N s for stations typical of each climatic division are
shown by figure 4.14.
Yet another facet of the climate is the year-to-year variation of the
monthly mean value of N s. Five consecutive years of monthly means
were prepared for each of the six typical stations whose annual cycles are
shown in figure 4.14. Then, for each month, the absolute value of the
difference between consecutive years was obtained. These values were
then averaged for all months and are listed in the second column of table
4.4.
104
N CLIMATOLOGY
TYPE I
MID-LATITUDE COASTAL
WASHINGTON, D.C.
TYPE II
SUBTROPICAL - SAVANNA
MIAMI, FLA.
TYPE m
MONSOON -SUDAN
JODPHUR, INDIA
TYPE EZ:
SEMI-ARID MOUNTAIN
DENVER, COLO.
TYPE Y TYPE 21
CONTINENTAL -POLAR ISOTHERMAL EQUATORIAL
OSLO, NORWAY CANTON ISL., S. PACIFIC 0.
JFMAMJJASONOJ JFMAMJJASONDJ JFMAMJJASONOJ
Figure 4.14. Representative annual cycles of Ns /or the major climatic types.
Another measure of the variation of monthly mean values of N ^ is
obtained by differencing the maximum and minimum values occurring for
a given month during the 5-year period. These differences are also given
in table 4.9 for the months of February, May, August, and November.
Table 4.4. Year-to-year differences of monthly mean Ns
Maximum difTerences between
monthly
Differences between monthly means
means over a 5-yr period
Climatic*
in successive years for the same
month, averaged for all seasons over
type
a 5-yr period
Feb.
May
Aug.
Nov.
I
5.7
6.0
16.5
17.0
7.0
n
5.4
8.5
6.5
8.0
11.0
III
8.9
16.5
14.5
20.0
6.5
IV_...
5.4
10.5
11.5
13.5
6.5
V
4.7
5.5
11.5
10.0
5.0
VI.
7.1
9.5
25.5
8.5
8.5
*Climatic types are the same as those in table 4.3.
APPLICATIONS
105
4.2.5. Applications
The communications engineer usually has available a small amount of
measured field-strength data from limited tests of a particular system.
He must then estimate the expected signal level or practical range of that
system, or other systems, for other tunes of the year, other years, and in
other areas. The variation of signal level from month to month and
climate to climate can be explained, in part, by its observed correlation
with Ns.
Pickard and Stetson [10, 11] were among the first to note the correlation
of Ns and received field strengths. The correlation of Ns and field
strength over a particular path has been studied quantitatively [12, 13]
and found to be highest (correlation coefficients of 0.8 to 0.95) when the
variables are averaged over periods of a week to a month. This latter
study has shown that the regression coefficient (decibel change in field
strength per unit change in N s) varies diurnally from 0.14 dB in the after-
noon hours to 0.24 dB per unit change of N s iu the early morning hours.
This correlation is so sufficiently consistent that Gray [14] and Norton [15]
have utilized it in their recent prediction methods of transmission loss in
a band from 100 to 50,000 Mc/s. In addition, the coefficient 0.2 dB per
unit change in A''^ has been tentatively adopted by CCIR Study Group V
in their revision of the 30- to 300-Mc/s tropospheric-wave propagation
curves to account for the geographic and seasonal variations of field
strengths. The estimates of field strength variations attributed to N s
given in table 4.5 are based upon the CCIR coefficient.
If one assumes, for comparison only, that the worldwide average value
of A^s is 330 and that one is able to estimate the field strength level of a
particular communications system at a given distance and for N s = 330,
then the above correlations would indicate that the climatic variations of
fields given in table 4.5 might be expected.
Table 4.5. Climatic variation of hypothetical communications system relative to pre-
dicted value for Ns = 330, assuming a 0.2-dB variation per unit change in Ns
Climatic type*
Expected yearly
mean field strength
level relative to
AT. =330
Expected annual
range on the above
assumption
I
II
Ill
IV
V... . .
dB
-6 to +4
+4 to +14
-10 to +14
-18 to -6
-6 to +2
+2 to +14
dB
6 to 12
6 to 12
12 to 20
Oto 12
to 6
VI
Oto 6
•Climatic types are the same as those in table 4.3.
106 N CLIMATOLOGY
The data of table 4.5 indicate, for example, that identically equipped
tropospheric communications systems could display as much as a 32-dB
difference in mean signal-strength level due to the climatic difference of
say, Denver, Colo., and the tropics. Further, one might expect the
monthly mean field strength of this hypothetical system to vary through-
out the year from less than 12 dB in the high plains near Denver to as
much as 20 dB in the African Sudan.
Under this same assumption, figures 4.10 and 4.11 allow the communi-
cations engineer to estimate the expected maximum and minimum
monthly mean field strength expected throughout the year.
The year-to-year variations of the monthly mean A''^ listed in table 4.4
indicate that the monthly mean of field strength for a particular month
may differ in successive years by as little as 1.0 dB for chmatic category
V in November or as much as 5.1 dB for category VI in May.
Another application of these worldwide charts is to aid in estimating the
refraction of radio waves. The most convenient method of accounting
for the effects of atmospheric refraction is by means of the effective-
earth's-radius concept (see chapter 3) of Schelleng, Burrows, and Ferrell
[16]. The effective earth's radius, Ug, is determined from
tte = I ;^V' (4-7)
I' + Ilj
where a is the true radius of the earth, n is the refractive index, and
dn/dh is the initial n gradient with respect to height. A great simplifica-
tion of propagation calculations is accomplished by assuming that dn/dh
is a constant, thus allowing radio rays to be drawn as straight rays over a
fictitious earth of radius a^ rather than curved rays over the true earth of
radius a. This simplification allows, for example, the distance to the
radio horizon, d, of a radio ray leaving an antenna of height, h, to be
calculated from d = ■\/2ae h.
One notes, however, that the determination of a^ involves dn/dh as
well as n and that our A^o charts allow only an estimation of n. This dis-
parity may be resolved by utilizing the observation that N s is highly
correlated with the value of A^ at 1 km above the surface. The difference
between A^^ and N at 1 km is denoted AN. It has been noted [9] that the
correlation coefficient between ^nJAA^j and Ns is 0.926 for 888 sets of
data from 45 U.S. weather stations representing many diverse climates.
The regression equation
-AiV = 7.32 exp {0.005577 N s} (4.8)
results when both variables are averaged over 6 to 8 years of record.
CRITICAL APPRAISAL OF RESULTS 107
Approximating dn/dh in (4.7) by AN, we may determine that the radio
horizon distance of an antenna located 150 m above the earth would vary
from 48 km when A''^ = 200 to 59 km when Ns = 400. Yet another ap-
plication of the N s charts is to the exponential models of the decrease of
refractive index with height which have been proposed to date [9, 17].
These models are completely specified by N s and may be used to account
for seasonal and geographic variations of such refraction effects as radar
range and elevation angle errors.
4.2.6. Critical Appraisal of Results
The world maps presented above were based upon data from 306
weather stations. This number of stations appears to be consistent with
the scale of map used. The map scale is so small, however, that only
large climatic differences can be expected to be discerned. For the
climate of any given area one should refer to detailed studies of N such
as those currently in preparation for the United States at the Central
Radio Propagation Laboratory.
The accuracy of the present charts may be assessed from the charts of
maximum range, R, of monthly means as given by figures 4.12 and 4.13.
The standard deviation of the individual monthly means may be esti-
mated from [18] 0.43 R, where the coefficient 0.43 is appropriate for five
individual observations. Since, in general, R < 20 N units, then
0.43 i? < 9N units, although this standard deviation may be as large as 26
N units for the month of February in Australia and 17 N units in the
southwest of the United States during August, or in the African Sudan
during February.
Further, the standard error of estimating a 5-year mean from five
individual monthly values is determined from
0A3R
_ )
\/n
where n for our case is 5 and thus the error of the 5-year mean would be
0.192 R. Remembering that R < 20 N units and assuming perfect skill
in drawing the contours, one would expect the standard error of estimate
to be less than 4 A^ units. This standard error can be as large as 12 A"
units in Australia where R = GO N units.
The value of A^s at each of the 20 test stations of table 4.2 was estimated
from the A^o contours with an rms error oi 5 N units which is consistent
with the standard error of estimate obtained from the range charts. In
the large areas of sparse data, such as the oceans and Russia, this un-
certainty rises to about 10 N units and thus the contours in these regions
are dashed.
108 N CLIMATOLOGY
At the time the present study was initiated it was felt that Ns should
be reduced to sea level by at least the dry term correction factor as in
(4.6). The absence of published work on models of A^ structure in the
free atmosphere encouraged the decision to rest on prudence and adopt
this dry term correction factor. Since that time several effective ex-
ponential models of the free atmosphere have been demonstrated [17, 9].
In future work a smaller value of H* on the order of 7.0, which corresponds
to the N decay in the free atmosphere, will be adopted. The adoption
of any value of H* between 6.5 and 7.5 km would have reduced the range
of A^'o values on figure 4.4 by no more than 2 A'' units. Since this reduction
in range is more than an order of magnitude less than the reduction of
(4.6) used to obtain No, it appears that the basic advantage of the method
has been realized.
A map of A^'o, such as that of figure 4.4, which represents a large con-
tinental area, may easily be compared and merged with maps for ocean
areas. This would be more difficult with N s, since, for example, the
strong gradient over California mainly represents the rapid altitude
variations of the Sierra Nevada Mountains. It has also been demon-
strated that A^'o is a better indicator of tropospheric storms and air
masses than N s when considered on a synoptic or "daily weather map"
basis [19].
Also (4.6) aids in comparing air properties as a function of altitude at
the same position. The variations of No will represent the local depar-
tures between this quantity and the value in a standard atmosphere and
will show the perturbations in the structure of the atmosphere produced
by fronts, air masses, and other synoptic features. Although any value of
H* between 6.5 and 7.5 km will remove the gross altitude dependence of
the refractive index, the choice of value of H* within this range could de-
pend on the application. The synoptic application, which is discussed in
detail in chapter 5, would be best served by a scale height near 8.0 km,
whereas the objectives of the climatic chart usage would best be met by
a scale height near 6.5 km. This seeming paradox is easily understood in
terms of the physical interpretation of the various scale heights. For
example, H* = 8.0 km is typical of a perfectly dry atmosphere and its use
results in emphasis of humidity differences between air masses, whereas
H* = 6.5 km corresponds to a saturated atmosphere and minimizes
moisture differences. Thus it would appear that eventually one might
use a value of H* as indicated by the application at hand. As a practical
matter, however, H* = 7.0 km appears to reach a desirable compromise
between the objectives of the two preceding examples.
4.2.7. Conclusions
With the above critical appraisal in mind, the salient conclusions of the
present study are :
DATA AND REDUCTION TECHNIQUES 109
(a) The radio refractive index varies in a systematic fashion with
climate and different climates may be identified by the range and mean
values of the refractive index.
(b) It is 4 or 5 times more accurate to estimate the station value of the
index from charts of the reduced-to-sea-level index than from charts of
the station value. This improved accuracy results from using a method
that allows height dependence to be accurately taken into account.
(c) Identically equipped tropospheric communications systems might
be expected to vary as much as 30 dB in monthly mean signal level in
different climatic regions, and the annual range of monthly mean field
strength could be as high as 20 dB in the Sudan of Africa and as low as
to 6 dB in the high plains of the western United States.
4.3. On the Average Atmospheric Radio Refractive
Index Structure Over North America
4.3.1. Introduction
As has been already noted, the radio refractive index of the atmosphere
combines three of the meteorological elements normally used to specify the
state of the atmosphere on either a synoptic or a climatological basis.
This fact has led to its being used as a synoptic tool [19, 20, 21, 22, 23] and
as a measure of climatic characteristics [7, 24, 25, 26, 27].
The present treatment is concerned with the degree to which the average
A^ structure in the vertical direction reflects the gross differences in climate
over the North American continent. Diurnal and seasonal range graphs
of N at the earth's surface also shed light upon climatic characteristics.
4.3.2. Meteorological Data and Reduction Techniques
The basic data used in this study are the significant level data of the
radiosonde observations from the 18 weather stations shown on figure 4.15
for the 5-year period 1952-57. These observations were converted by
means of (1.20) to radio refractivity, N.
The significant level data were collected for the values of N at the
earth's surface and within height increments centered upon 0.25, 0.5, 0.75,
1.0, 1.5, 2.0, 2.5, 3.0, and 3.5 km. Each value of N was referenced to the
center of its height increment by use of the average atmosphere,
N{h) = Aoexp [-h/1.0\, (4.9)
which has been shown to be a reasonable model for the decrease of the
refractive index with height for the United States. By use of this ex-
ponential model, the gross height dependence of N within each height
no
N CLIMATOLOGY
Figure 4.15. Radiosonde stations used in this study.
increment was effectively removed and there remains a more reliable
estimate of the mean N at the center of the height increment. The final
value of A^'o, for example, at 1 km over Washington, D.C., would be 300
with a standard deviation of 15.7 N units for 152 observations. The
error in the mean is then determined to be 1.3 from
s(N) =
s(N)
Vk
(4.10)
with k = 152. An examination of similar data for all stations and levels
used shows that s{N) is generally less than 1.0 A" unit.
In addition, when considering atmospheric cross sections, the mean
values of A were referred to sea-level by means of (4.9) thus further em-
phasizing climatic differences [28].
AVERAGE A^o STRUCTURE 111
4.3.3. Average N„ Structure
The first series of charts presented in this analysis are those showing
the time variation of A^'o both vertically above each station as well as at
the standard ground-observing level. A slight dichotomy in data sample
exists since the vertical data are monthly averages of radiosonde observa-
tions taken twice a day, corresponding in time to the hours 0300 and 1500
GMT, while the surface data are 8-year means for the even hours of the
day local time [29]. This climatic variation at each station is represented
by a two-part chart prepared for each station, the first part depicting
seasonal changes in the mean value of No throughout the first 3.5 km
above the station and the second showing the seasonal and diurnal changes
of No at station elevation. These two presentations give a more complete
climatic picture of a location than do the usual unidimensional annual
cycle graphs.
Tatoosh Island (TTI), figure 4.16, off the coast of Washington State,
illustrates typical features of a marine west coast climate. The seasonal
profile changes indicate moderate gradients the year around with a small
summer maximum in the No gradients in keeping with the cool tempera-
ture regime and small diurnal temperature range of this maritime-
dominated climate. This consistency of weather conditions is further
emphasized by the almost complete lack of diurnal pattern in the surface
data throughout the year and small seasonal change in No for any hour
of the day.
The seasonal profile chart of Oakland (OAK), figure 4.17, indicates less
maritime effect than at Tatoosh Island and shows clearly the influence of
the summer subsidence inversion on the northeastern edge of the Pacific
high-pressure area by both the low values of No and increased gradients
at about 1.0 km. The surface data show the moderate seasonal and
diural cycles for all months of the year expected in a Mediterranean
climate.
The effects of the Pacific High are more pronounced further south along
the California coast. The vertical profile data for San Diego (SAN),
figure 4.18, show clearly that during the winter months the strongest
gradients of A^o are near the surface as one would expect in a maritime
dominated climate while, by contrast, in the summer the well-known
Southern California elevated inversion produces strong gradients aloft.
The subsiding air above the inversion is reflected by a characteristic in-
crease of No with height. The maritime effect is also evident at the
surface in the low diurnal range of about 10 N units. It is significant
that the presence of the elevated layer produces a smaller diurnal range
in the summer than in the winter.
The semitropical nature of the humid periphery of the Gulf of Mexico
is reflected by the data for Brownsville, Tex. (BRO), figure 4.19, which
show a strong seasonal cycle and pronounced diurnal range in the summer
112
N CLIMATOLOGY
Figure 4.16. Diurnal, seasonal, and vertical variation of No for Tatoosh Island,
Wash. (TTI).
due in large measure to the very high summer temperatures in this locale
with resultant high water-vapor capacity. It is also quite striking that
the general level of A^o is 25 at 30 iV" units above that of the west coast.
The high A^o values of the Gulf regions are also found at Miami (MIA),
figure 4.20, although the moderating effect of continual onshore winds
produces less pronounced vertical gradients and smaller diurnal ranges
than Brownsville, particularly during the summer months.
AVERAGE iVo STRUCTURE
113
Figure 4.17. Diurnal, seasonal, and vertical variation of No/or Oakland, Calif. (OAK).
The interplay of polar and maritime air masses along the middle east
coast of the United States is reflected in the strong seasonal range of 50 A^^
units at Washington, D.C. (DCA), figure 4.21. The summertime diurnal
range of 15 A" units reflects the moderating influence of maritime air so
common along the central east coast in the summer.
The latitudinally controlled lower mean temperatures of the northeast
coastal regions are reflected in the generally lower values of A'^o plus
114
N CLIMATOLOGY
Figure 4.18. Diurnal, seasonal, and vertical variation 0/ No/or San Diego, Calif. (SAN).
smaller seasonal, diurnal, and vertical ranges as illustrated by Portland,
Me. (PWM), figure 4.22. The curious low in late autumn appears to be
the result of advection of air from the continental interior and is perhaps
indicative of the Indian summer of New England. The long New Eng-
land winter appears in the surface data as a large area of nearly constant
No.
The above data reflect the south-to-north change from humid sub-
tropical to marine-modified continental climates typical of the lee coasts
AVERAGE No STRUCTURE
115
Figure 4.19. Diurnal, seasonal, and vertical variation 0/ No/or Brownsville, Tex. (BRO).
of continents. By contrast, the island-like station of Cape Hatteras
(HAT), figure 4.23, reflects both the characteristic strong seasonal range
of a lee coast station that is dominated by dry, cold continental air in the
winter and warm, moist maritime air in the summer and the small diurnal
ranges for all months of the year of a maritime modified climate.
The data for Denver, Colo. (DEN), figure 4.24, strongly reflect the
influence of the climatic controls of altitude and continentality. The level
of A''o is intermediate to that of Oakland and Washington. The vertical
116
A^ CLIMATOLOGY
Figure 4.20. Diurnal, seasonal, and vertical variation of No /or Miami, Fla. {Ml A).
profile chart for this station indicates the mild influx of tropical air in the
summer at the surface and the convective mixing of this air to great heights
by the strong thunderstorm activity common to the area. The high No
values between 2 and 3 km are perhaps due to superior air subsiding on
the lee slopes of the Rocky Mountains. Relatively intense diurnal ranges
are apparent in both summer and winter, as the day to night variations of
this high plains climate are controlled to a large extent by radiational heat-
ing and cooling through the thin atmospheric blanket at high elevations.
AVERAGE No STRUCTURE
117
Figure 4.21. Diurnal, seasonal, and vertical variation 0/ No for Washington, D.C.
(DCA).
The strong warm-season influx of tropical maritime air from the Gulf
of Mexico up the Mississippi Valley is spectacularly in evidence at
Columbia, Mo. (CBI), figure 4.25. The most significant feature of these
charts is the weak year-round diurnal range of 10 A^ units or less accom-
panied by a strong seasonal range of nearly 50 N units.
In summary, it is seen that the west-coast stations display a rather
uniform low average value of A^o accompanied by small seasonal and
diurnal ranges. This is due to the continual onshore advection of cool
118
N CLIMATOLOGY
Figure 4.22. Diurnal, seasojial, and vertical variation ofNofor Portland, Me. (PWM).
maritime air that keeps the mean temperatures both low and uniform.
By comparison, the continental stations show relatively large diurnal and
seasonal ranges controlled in large measure by the radiative heating and
cooling, summer to winter and day to night. The east-coast stations
display a general increase of iVo from north to south arising from the
AVERAGE No STRUCTURE
119
Figure 4.23. Diurnal, seasonal, and vertical variation of No /or Cape Hatieras, N.C.
{HAT).
general increase of mean temperature with resultant increase in water-
vapor capacity. These same stations have a marked seasonal range due
to the interplay of continental and maritime effects. The relatively
small summertime diurnal ranges along the east coast reflect the strong
influence of maritime air upon this region during that season.
120
N CLIMATOLOGY
Figure 4.24. Diurnal, seasonal, and vertical variation of No /or Denver, Colo. (DEN).
AVERAGE No STRUCTURE
121
Figure 4.25. Diurnal, seasonal, and vertical variation of No for Columbia, Mo. (CBI).
122
N CLIMATOLOGY
4.3.4. Continental Cross Sections
The vertical profile data obtained for the stations of figure 4.15 provide
a means to construct A'^o cross sections traversing varied climatic zones
and geographic regions.
The first cross section of this series, figure 4.26, is taken along the
Pacific coast of central North America from Canada to Mexico. Intru-
sion of polar air at the northern end of the wintertime cross section is
responsible for the relatively flat N gradient over Tatoosh. The 320
A^'o-isopleth covering most of the coast southward from Tatoosh shows the
uniform modifying influence of the ocean which, at this time of year, is
considerably warmer than the continent. Over southern California the
low at 500 m results from the drying-out effects of the Pacific high inver-
sion. On the summer cross section moderate A'"o-values typify the coast.
The minimum at Oakland stems perhaps from the upwelling of relatively
cool ocean waters off the California coast. Striking evidence of the
Pacific inversion is apparent from the low at 1 km between Oakland and
San Diego that results from significant decreases in the vapor pressure
term contribution to N within the dry subsiding air aloft.
SAN TTI
2
DISTANCE (KmxIO')
Figure 4.26. North-south No cross section along the western coast of the United States.
CONTINENTAL CROSS SECTIONS
123
FEBRUARY
^ >25
335
I AUGUST I
PWM
OCA HAT
COF PWM
2
DISTANCE (KmxIO')
Figure 4.27. North-soxdh No cross section along the eastern coast of the United States.
The cross section for the eastern seaboard of the United States, figure
4.27, presents a considerably different picture of refractive index chmate
than the Pacific coast. On the winter cross section the area between
Portland and Washington falls within a low produced by frequent occur-
rence of polar air masses during this season of the year. Southward to-
ward Cocoa, Fla. (COF) there is a significant A'' increase that corresponds
to the considerable latitudinal temperature gradient that exists on the
east coast of the continent during the winter months. On the summer-
time cross section the temperature gradient difference between the two
coasts is even more pronounced and is reflected in the strong east-coast A^'o
difference of about 40 N units as compared to the Tatoosh-San Diego
difference of about 10 A'' units. The vertical gradients are also seen to
increase in a systematic fashion from north to south, reflecting the in-
creased vapor pressure gradients in the warmer southern regions.
The extensive continental cross section from Isachsen (IC) to Balboa
(BLB), figure 4.28, discloses several interesting features. On the winter-
time chart the relatively high A^o values between Isachsen and Churchill
(YQ) indicate the presence of very cold, dense, dry air. To the south over
Bismarck, N.Dak. (BIS), and the Great Plains, relatively warm, dry air of
124
A^ CLIMATOLOGY
YQ BIS CBI BRO
234567
DISTANCE (KmxIO
Figure 4.28. North-south No cross section from Isachsen, North West Territories (IC).
through Churchill (YQ) to Balboa, Canal Zone (BLB).
CONTINENTAL CROSS SECTIONS
125
FEBRUARY I
325 \ V J '
\
/ •'530\ \
/ / ^ ^
AUGUST
TTI
3 4
DISTANCE (KmxIO'
Figure 4.29. Wesi-eoM No cross section for the northern United States.
lower density creates a refractive index low. Southward from the
Great Plains a considerable A^'o gradient is encountered as the Gulf source
region for tropical maritime air is approached. Latitudinal temperature
gradients produce a continued increase in A^'o to the southern extreme of
the chart at Balboa in the tropics. The A^o gradients of the summer cross
section are largely thermally controlled, ranging from cool temperatures
and low A'^o in the polar regions to a maximum between Brownsville and
Balboa near the warm-season heat equator.
Three zonal cross sections have been prepared extending across the
northern, central, and southern portions of the United States. The
February chart for the northern cross section, Tatoosh to Portland, figure
4.29, exhibits lows at both coasts, with slightly higher values of A^o iu the
cold interior of the continent. The summertime chart clearly shows the
intrusion of tropical maritime air from the Gulf of Mexico pushing north-
ward over the Great Plains. The central U.S. cross section, figure 4.30,
shows, on the wintertime chart, slightly lower values of A^o on the west
coast at Oakland than at Hatteras. East of the Rockies, the chinook
winds of the prevailing westerlies assist in the formation of a relatively
warm, dry air mass of characteristically low A^o- On the summer map,
126
N CLIMATOLOGY
I FEBRUARY
330
I AUGUST
DEN CBI HAT OAK
2 3 4
DISTANCE (KmxIO')
Figure 4.30. West-east No cross section for the central United States.
the influence of the Pacific high is in evidence over Oakland, while in the
interior of the continent the influence of tropical maritime air is noticeable
up the Mississippi Valley. The zonal cross section across the southern
United States, figure 4.31, shows a transition from the Mediterranean
climate of the Pacific coast to the humid subtropical type from Browns-
ville eastward. Here again the east-coast A^o values, from a higher
temperature regime, are larger than those in the west. The summer-
time chart is similar to the winter one with the A^-gradient intensified all
along the route.
Figure 4.32 is for the longest cross section, extending approximately
22,000 km from Canton Island (CAN) to Wiesbaden (WSB) in northern
Europe. The winter chart shows a gradual decrease in N from Canton
Island to Brownsville and Cocoa, with very uniform A^ structure over the
Atlantic to Ship "J," and onwards to Wiesbaden. The summer chart
shows a double maximum, one at Canton and the second over the warm
North American continent between Brownsville and Cocoa. The gradual
decrease in mean temperature northward across the Atlantic Ocean is
evidenced by the rather uniform decrease of A^o between Cocoa and
Wiesbaden.
CONTINENTAL CROSS SECTIONS
127
Figure 4.31. West-east No cross section for the southern United States.
128
N CLIMATOLOGY
3.0
'325
CAN
BRO COF .r
10 '^3
DISTANCE (KmxIO )
FEBRUARY
I AUGUST I
"J" WSB
20 25
Figure 4.32. West-east No cross section from Canton Island, South Pacific (CTN)
through Wiesbaden, Germamj (WSB).
CLIMATIC CHARACTERICS 129
4.3.5. Delineation of Climatic Characteristics
Since climate is really a synthesis of weather elements taken over a
period of time, it is apparent that no fixed climatic boundaries exist in
nature. These boundaries shift from year to year with changing weather
and with the addition of new data into the climatic averages. Climatic
borders, then, represent transition zones between so-called "core climates"
of one type or another. A "core climate" presumably maintains its
climatic personality consistently over a period of time. The coniferous
forest regions of the far north of Canada, for example, maintain a con-
tinuously frigid climate under the arctic inversion during the heart of the
winter season. Since a simple system is required for effective classifica-
tion, the congeries of minute climatic areas arising from extensive climatic
division are, for practical purposes, coordinated into areas having broad
similarity of climatic character. Thus we shall only attempt here to look
for general patterns in our limited data sample.
The A^'o data for the station elevation of figures 4.16 to 4.25 contain
information about the radio refractive index climate of the respective
stations that may be converted to a pair of indices, one seasonal, one
diurnal, useful in classifying climate. The diurnal index is the ratio
obtained by dividing the difference between the highest and lowest hourly
means for August by a similar difference for February. This ratio is then
plotted versus a ratio of the maximum difference for the 12 mean A'^o
values at 0200 divided by the maxmium differences for the 1400 means.
The results of this analysis can be seen on figure 4.33, where the largest
pair of ratios are observed for Denver and Bismarck, a consequence of
the strong continentality effects at these inland stations. Denver, it is
to be noted, displays a large seasonal ratio whereas Bismarck represents
the extreme in diurnal ratios. Brownsville, Washington, and Columbia
show sizable seasonal ratios coupled with slightly greater than normal
diurnal ratios. Summer-to-winter climate changes are considerable at
these three locations.
The remainder of the stations display the near-unity value of both
ratios that would be expected of maritime-dominated climates.
A comprehensive, though simple, index of climatic variation of the
vertical distribution of A'' is particularly difficult to envisage. One such
index is the ratio, R{AN), of the summer to winter value of AA^ versus the
mean value of AA^ (the absolute difference in A^ at the earth's surface and
at 1 km above the earth's surface). The gradient, AA'', has received wide
engineering application and is currently being mapped on a worldwide
basis by the International Consultative Committee for Radio [30]. When
this ratio is plotted versus the yearly mean value of AA^, as on figure 4.34,
one again obtains about the same climatic demarcation as above. For
example, the maritime-dominated climates generally have values of
RiAN) of 1.2 or less.
130
N CLIMATOLOGY
2.0^
®
DEN
1.8-
C=3
fee
Cd
1.6-
«a:
•r^
CO
1.4-
®
-3Z
U-J
BRO
OO
1.2-
®DCA
^^
-a:
1.0_
(
CAM HAT PU/U® ^^'
SAN® (^MiA PWM^
COF® ®
'" OAK
®BIS
1 1 1 1 1
) 1,0 2,0
3,0
1 1
4,0
MEAN DIURNAL RATIO
Figure 4.33. Comparison of mean seasonal to mean diurnal ratio of No at the earth's
surface.
The more southerly and humid stations have relatively high mean AA^
values in excess of 55 A'^ units/km. The single European station, Wies-
baden, shows both low average value of AA'^ arising from the low average
temperatures and small seasonal variation associated with the continuous
onshore advection of cool, moist air from the Atlantic Ocean. The values
R{AN) < 1.0 for the Canadian stations of Churchill and Isachsen reflect
the steep gradients associated with the intense stratification during the
long winter night. The high, dry climate of Denver produces low
mean values of AA^ but a high seasonal range. The interplay of conti-
nental and maritime effects along the east coast produces relatively high
values of R{AN) while the presence of the Pacific high inversion layer
below 1 km accounts for similar values along the west coast. The ex-
tremely high value of R{AN) of 1.8 for Tripoli (TPI) arises principally
from the intense summertime A^ gradients that exist near the surface in
CLIMATIC CHARACTERISTICS
131
2.0j
■^
^
1
®
ce:
1.8.
TPI
<S
^
1.6.
CXL
^
1.4.
I.2_
DCA ®^^' ^ ®HAT
U-]
en
-a:
en
C_3
^BIS ® OAK®
T '" CTN
*£
MIA® ^®^'^°
®WSB ""^ OOF
cz>
l.0_
CO
®YQ n^n
U_J
BLB
OO
.8_
3
®IC
40 50 60 TO
YEARLY MEAN AN
Figure 4.34. Yearly mean and range ratio of AN.
this locale. In the lowest layers of the atmosphere warm, moist air from
the Mediterranean Sea is advected inland underneath superheated air
from the Sahara Desert. Along the zone of contact between these air
masses of widely differing moisture content, spectacular drops in vapor
pressure and, consequently, in the wet term component of A'^ occur.
During the winter, moderate temperatures and prevailing offshore flow
combine to produce near-normal gradients of N. Consequently the
summer-to-winter AiV ratio turns out to be quite high for this borderland
station.
The analysis above represents a rough and crude first attempt to under-
stand the climatic variations of N . It is expected that such classifica-
tions will improve with the addition of data from more diverse locations
and, in particular, as our knowledge of the role of water vapor in climatic
variability increases.
132 A^ CLIMATOLOGY
4.4. The Climatology of Ground-Based Radio Ducts
4.4.1. Introduction
The occurrence of atmospheric ducts places limitations upon ray tracing
of VHF-UHF radio waves. Ducting is defined as occurring when a radio
ray originating at the earth's surface is sufficiently refracted so that it
either is bent back towards the earth's surface or travels in a path parallel
to the earth's surface. Although the rigorous treatment of ducting in-
volves consideration of the full wave eciuation solution [31] rather than a
simple ray treatment, the present study will be based upon a geometrical
optics definition of the limiting case in which ray tracing techniques may
be used. This simple criterion is then applied to several years of radio-
sonde observations from stations typical of arctic, temperate, and tropical
climates to derive estimates of the variation of the occurrence of radio
ducts with climatic conditions.
4.4.2. Meteorological Conditions Associated With Radio
Refractive Index Profiles
The path followed by a radio ray in the atmosphere is dependent upon
the gradient of the refractive index along that path. Of the vertical and
horizontal gradient components that compose the path gradient, the
horizontal gradient is normally negligibly small. Thus, the atmosphere
is considered horizontally homogeneous and only the vertical gradient of
the refractive index is utilized. The numerical value of the vertical
gradient of the index of refraction depends on the vertical distribution
of atmospheric temperature, humidity, and pressure.
Normally, temperature and humidity decrease with height in the atmos-
phere, since turbulence prevents any great changes in structure. How-
ever, there are periods of time in which the air becomes fairly calm,
whereupon temperature inversions and humidity lapses can be built up
and maintained. Temperature inversions have a twofold importance in
that (a) they can be widespread in area and persist over a relatively long
period of time, and (b) they exercise a stabilizing influence on air motion
such that turbulence is suppressed and strong humidity gradients may
develop. Layers in which there is intense superrefraction to the point of
duct formation may be formed as a result of these gradients and trapping
of radio waves may follow. The inversions may start at ground level or
at some greater height. The thickness of the layer can show great
variability. Three processes that form temperature inversions are:
Advection: Advection is the horizontal flow of air having different heat
properties. Such a process is of importance in microwave propagation
since it may lead to a different rate of exchange of heat and moisture be-
tween the air and the underlying ground or ocean surface, thus affecting
RADIO REFRACTIVE INDEX PROFILES 133
the physical structure of the lowest layers of the atmosphere. This proc-
ess results in air with different refractive index characteristics being
brought into the area. The most common and important case of advec-
tion is that of dry air above a warm land surface flowing out over a cold
sea. This type of advection frequently appears in the English Channel
during the summer when the weather has been fine for several days.
Advective duct formation depends on two quantities: (a) the excess of
the unmodified air temperature above that of the water surface, and (b)
the humidity deficit (the difference between the water vapor pressures of
the modified and unmodified air). If these quantities are large, especially
the humidity deficit, an intensive duct may form.
Important advective processes can also occur over land, but the condi-
tions required for duct formation occur less frequently. However, a duct
may be formed when dry, warm air flows over cold, wet ground with
resultant temperature and humidity structure previously discussed.
Radiation: Differences in daytime and nighttime radiation are the
causes of diurnal variation in refractive conditions. A subref ractive layer
may be present during the day, especially at the time of maximum surface
heating. Clear skies and light surface winds at night result in consider-
able cooling of the earth, thus causing the formation of temperature in-
versions. The surface heat loss produced by nocturnal radiation is a
prime factor in the formation of temperature inversions. Atmospheric
stratification formed by such a combination of meteorological parameters
may jjroduce a trapping layer. A temperature inversion is seldom strong
enough to produce a duct in the middle and low latitudes, but it is of
major importance in the formation of ducts in the northern latitudes.
Low stratus clouds or extreme amounts of moisture (as in the troj^ics)
tend to prevent loss by radiation which lessens the possibility of duct
formation.
If the ground temperature in a nocturnal inversion falls below the
dewpoint temperature, the water vapor in the lowest layers of the air
condenses and the heat of condensation is released directly to the air.
Under conditions of radiative fog formation, the humidity lapse tends to
counteract any temperature inversion present and may cause substandard
refraction if the humidity "inversion" is sufficiently strong. However,
the temperature inversion may be strong enough to keep the layer
standard or superrefractive.
Subsidence: Subsidence is the slow settling of air from a high-pressure
system. The air is heated by adiabatic compression as it descends and
spreads out in a layer well above the earth's surface. This process pro-
duces stable layers and inversions of temperature with an accompanying
decrease in relative humidity. Since the air has come from a high level
in the atmosphere, it is dry and may overlay a cooler, moist air mass.
134 A^ CLIMATOLOGY
This type of inversion may cause the formation of an elevated super-
refractive layer where the air temperature usually decreases immediately
above the ground, rises through the inversion layer and decreases above
the layer. This is a common occurrence which may be observed at any
time. Subsidence has a tendency to destroy subrefractive layers and to
intensify superrefractive layers. Although the effects of subsidence are
generally observed at high levels, they are occasionally observed at lower
levels, especially in the subtropics. Since subsidence frequently occurs
in the lee of mountains and in the southeastern regions of northern
hemisphere highs, elevated ducts may also be observed.
Conditions inimical to ducting are those which mduce mixing of the
lower atmosphere. Small scale atmospheric motions (turbulence) and
consequent mixing and mass exchange result from differential surface
heating and surface roughness effects. Both of these processes work to
destroy stratification. They ultimately result in uniform vertical distri-
bution of moisture through considerable depths of the lower atmosphere
and the establishment of neutral temperature lapse rates. Accordingly,
where the process of mechanical and convectively induced mixing are at
work, the probability of the occurrence of ducting is vanishingly small.
Thus, few, if any, ducts are observed over snow-free, low albedo land
areas from midmorning to late afternoon when the skies are clear, or in
areas of moderate to great surface roughness when the surface winds are
more than a few meters per second, irrespective of cloud conditions or the
time of the day.
4.4.3. Refractive Conditions Due to Local
Meteorological Phenomena
Land and sea breezes may produce ducts along the coastal regions since
the winds are of thermal origin, resulting from temperature differences
between land and sea surfaces. During the day, when the land gets
warmer than the sea, the air above the land rises and is replaced by air
from the sea, thus creating a circulation from the sea to the land, called a
sea breeze. During the night, the land becomes colder than the sea and a
circulation, called a land breeze, is set up in the opposite direction. This
type of circulation is generally shallow and does not extend higher than
a few hundred meters above the land or sea surface.
A land or sea breeze may modify the refractive conditions in different
ways depending upon the distribution of moisture in the lower layers.
Since these breezes are of a local nature and generally extend only a few
miles, only coastal locations are usually affected. The very nature of sea
and land breezes results in a marked refractive index pattern. With a
BACKGROUND 135
sea breeze a duct may be formed over the water due to subsidence. The
land breeze is accompanied by subsiding air over the land with resultant
duct formation.
The formation of fog results in a decrease of superrefractive or ducting
possibilities. When a fog forms by nocturnal radiation, the water content
of the air remains practically the same; however, part of the water
changes from the gaseous to the liquid phase thus reducing the vapor
pressure. The resulting humidity lapse rate tends to counteract the
temperature inversion and cause above standard refraction. However,
the temperature inversion may be strong enough to keep the layer stand-
ard or superrefractive. This process may also occur with advection fogs.
The nighttime temperature profile is a result of the interaction between
nocturnal radiation, turbulence, and heat conduction. The associated
refractive index profiles are such that a radar duct begins to form about
the time of sunset, developing quickly during the early evening, more
slowly after midnight, and dissipating rapidly after sunrise. This is
mainly an inland effect resulting from large diurnal temperature varia-
tions observed in the interiors of large continents. However, a shallow
body of water may have an appreciable diurnal temperature variation, as
compared to the open ocean so that superrefraction may occur over such
a location from time to time.
It is generally recognized that radiosonde observations (RAOB's) do
not have a sufficiently high degree of accuracy to be completely acceptable
for use in observing changes in the degree of stratification of the very
lowest layers of the atmosphere; however, until more accurate methods
such as meteorological towers and refractometer measurements are more
commonly used, the RAOB will continue to be used as a basis for fore-
casting the occurrence of superrefractive conditions.
4.4.4. Background
The property of the atmosphere basic to radio ray tracing is the radio
refractive index of the atmosphere, n, which for VHF-UHF frequencies
at standard conditions near the surface, is a number of the order of 1.0003.
Although the refractive index is used in ray tracing theory, it is more
convenient when evaluating refraction effects from common meteoro-
logical observations to use the refractivity A'', which, for the frequency
range to 30,000 Mc/s, is given by (1.20).
When evaluating the meteorological conditions that give rise to refrac-
tive phenomena, it is frequently instructive to examine separately the
behavior of the dry component, D, and wet component, W, of N .
136
A^ CLIMATOLOGY
The gradient of the refractivity, AA'', with respect to height may then
be expressed :
AN = AD -\- AW.
(4.ii:
Average values of AA^, AD, and AW are given in table 4.6 for two incre-
ments between the earth's surface and 1 km above sea level for Fairbanks,
Alaska; Washington, D.C.; and Swan Island, W.I.
Table 4.6. Gradient of N, D, and W in N units per kilometer
Station
Height
increment
February
August
-AZST
-^D
-hW
-^N
-AZ)
-^W
Fairbanks, Alaska
Washington, D.C
Swan Island, W.I
surface— 0. 5 km
0. 5km-1.0km
surface— 0. 5 km
0. 5km-1.0km
surface— 0.5 km
0. 5km-1.0km
37
35
41
30
39
58
41
35
34
26
24
24
-4
7
4
15
34
31
36
60
46
47
66
27
24
28
24
26
24
4
12
32
22
21
42
Several general observations may be made of the data of table 4.6:
The gradient of the dry term is relatively less variable than that of the
wet term when considered as a function of season or height; the increase
of AA^ from winter to summer at a particular location or from arctic to
tropical climate at a given time is most strongly reflected in AW rather
than in AD. The marked increase of gradient with height for Swan Island
reflects the drop of refractivity across the interface of the trade wind
inversion where dry subsiding air overlies the moist oceanic surface layer.
A fundamental equation used in radio ray tracing is Snell's law, which,
for polar coordinates, is given in chapter 3 as
nr cos 6 = rioVo cos do,
(4.12)
where n is the radio refractive index of the atmosphere, r is the radial
distance from the center of the earth to the point under consideration, and
6 is the elevation angle made by the ray at the point under consideration
with the tangent to the circle of radius r passing through that point. The
radius to any point, r, is equal to (a + h), where a is the radius of the
earth and h is the height of the point above sea level. The zero subscript
refers to the value of n, r, or d at the earth's surface.
For the present study, geometrical optics techniques, similar to those
considered by Bremmer [32], are used to indicate when refraction is
BACKGROUND 137
sufficiently great to direct a ray either back to earth or in a circular path
at a constant height above the earth; i.e.:
noVo cos do ^ ,
nr ~ '
This condition then allows one to obtain the value of do which divides the
rays into two groups: those that penetrate the duct and those that are
trapped within the duct. This particular value of ^o, called the angle of
penetration and designated dp, is obtained by noting those instances where
Mo > 1 (414)
nr
and solving for the value of ^o such that (4.13) is equal to unity.
It is instructive to consider the order of magnitude of refractive index
gradient needed for trapping for several commonly observed refractive
index profiles. If we rewrite Snell's law,
ritrt cos dt = HaTa cos dd, (4.15)
where the subscripts t and A refer to the values of the variables at the
transmitter height and the top of the trapping layer, then trapping
occurs when
rit Tt
riA Va
> 1, (4.16)
and the angle of penetration at the transmitter, dp, is given by setting
cos dp = 1. (4.17)
nt Vt
riA Ta
The maximum permissible n gradient for a given value of dp is then given
by:
^ ^ ru^^UA ^
t^r ta — Tt
where ua must satisfy (4.17); i.e.:
_ Villi cos d^. (4.19)
Ta
138 N CLniATOLOGY
By designating Va = rt -\- /1.4, (4.18) becomes
An
Ar
1 -
rt + hA
cos Or,
(4.20)
By rewriting (4.20)
An
rit
hA
1 -
1
Ar
1+^
rt
cos dr
(4.21)
and expanding (1 + hA/rt)~'^ and cos Op, one obtains the expression:
Lr,
An _ 1^1
Ar - ~ ''' Irt + 2hAA
(4.22)
by neglecting terms 0p/4! and QiA/rtY, and terms beyond these. For
the case of Gp = 0, (4.22) reduces to :
An _ n 1
Ar a a
157 N units/km.
(4.23)
It is seen from (4.22) that the n gradient necessary to trap a radio ray
at a given value of dp is practically independent of transmitting antenna
height above the earth. For example, a, 6p = ray will be trapped by
an n gradient of —157.0 A'' units/km at sea level, where Ut ~ 1.0003,
while the necessary n gradient at 3 km above sea level will be —156.9
A^ units/km for a,n Ut = 1.0002. This indicates, for all practical appli-
cations, that the necessary n gradient for trapping is independent of
altitude. Further, by considering the temperature and humidity gra-
dients encountered in the troposphere, one is led to the conclusion that
ducting gradients would not be expected to occur at altitudes greater
than 3 km. In fact, Cowan's [33] investigation indicates that trapping
gradients are nearly always confined to the first kilometer above the
surface.
A consideration of (4.22) indicates that the magnitude of the negative
gradient necessary for ducting is 1/a for dp = but increases in propor-
tion to dp'/2h as dp increases. The gradients necessary for atmospheric
ducts as a function of 60 are given for several different profiles in figure
4.35 An analysis of radiosonde data indicates that gradients in excess of
0.5 N units per meter are seldom exceeded within atmospheric layers.
It is interesting to note how rapidly the necessary gradients increase to
BACKGROUND
139
1300
1200
1100
1000
0^ 900
800
2 X
700
600
500
400
300
200
100
/
/
/
/"
K
^
Ele
O.Skr
and
voted Lo)
7 above g
/km tfiic
320, Nas
/er~~-__^
round
Ir
^^^
yj
No =
= 300
^
1
,
'
/
/
1'
'pproximc
adio sond
ite upper
e-observ
limit of
edgrodlet
Its. /
'
/
^
^
Grour
0^
d Based
03 km th
Nq = 320
.oyer — —
/
^
x;
Ground Be
ised Loye
^
^0-
320
_-
— -^
limum Gr
/a
4 5 6
Br, in Milliradians
10
Figure 4.35. Refradiviiy gradients needed for radio ducts.
the approximate upper limit of radiosonde-observed gradients: a ground-
based layer 100 m thick attains this gradient at 8.3 mrad while the
maximum observed gradient is intercepted by the 30-m layer curve at
^0 = 4.5 mrad. A third example was calculated for an elevated layer 0.5
km above the ground and 100 m thick by assuming normal refraction
between the ground and the base of the layer and solving for the necessary
ducting gradient within the layer. The large values of n gradient neces-
sary for this case explain why elevated ducts were not observed. Al-
though the preceding examples were calculated for a ground transmitter,
the combinations of dp, An/Ar, and Ar are very nearly the same as would
be obtained for any other transmitter height within the first 3 km above
the surface.
140 N CLIMATOLOGY
4.4.5. Description of Observed Ground- Based
Atmospheric Ducts
Approximately three years of radiosonde data typical of an arctic
climate (Fairbanks, Alaska), a temperate climate (Washington, B.C.),
and a tropical maritime climate (Swan Island, W.I.) were examined by
means of a digital computer for the occurrence of ducts during the months
of February, May, August, and November. The percentage occurrence
of ducts is shown on figure 4.36. The maximum occurrences of 13.8
percent for August at Swan Island and 9.2 percent for Fairbanks in
February are significantly greater than the values observed at other times
of the year. The Washington data display a summertime maximum of
4.6 percent. These data indicate that the temperate zone maximum
incidence is about one-half the wintertime maximum incidence in the
arctic, and about one-third of the summertime tropical maximum.
The range of observed values of 0p is shown in figure 4.37. The mean
value calculated for each month as well as the maximum and minimum
values of dp observed for the limiting cases are given for each month and
location. The mean value of the angle of penetration under these condi-
tions is between 2 and 3 mrad and appears to be independent of climate.
The maximum value of dp observed during ducting is 5.8 mrad.
The refractivity gradients observed during ducting are given on figure
4.38. The maximum gradient of 420 N units per kilometer was observed
during February at Fairbanks, Alaska. The mean values of N gradient
appear to follow a slight climatic trend from a high value of 230 N units
per kilometer at Fairbanks to a value of 190 A^ units per kilometer at
Swan Island.
Another property of radio ducts is their thickness, which is given in
figure 4.39. Again there is observed a slight climatic trend as the median
thickness increases from 66 m at Fairbanks to 106 m at Swan Island.
These values of thickness correspond to the gradients given in figure 4.38.
One can then obtain, by linear extrapolation, the thickness at which the
gradient is equal to — 1/a; i.e., the height corresponding to the gradient
just sufficient to trap the ray at do = 0. These values, shown on figure
4.40, display an increase in the median thickness of about 25 percent for
Swan Island, 100 percent for Washington, and 200 percent for Fairbanks,
which results in a reversal of the climatic trend of the observed thickness
between Fairbanks and Swan Island. This increase in height emphasized
the preceding conclusion. Fairbanks is characterized by shallow layers
with relatively intense gradients.
These maximum duct widths may be used to estimate the maximum
radio wavelengths trapped. Kerr [34] gives the maximum wavelength,
X, trapped by given thickness, d:
X„.ax = cy'" d''\ (4.24)
GROUND-BASED ATMOSPHERIC DUCTS
141
16
15
14
en
u
1.^
=3
Q
"D
12
CD
cn
O
CD
1 1
"O
c
13
lU
O
V
CD
9
O
c
CD
U
\_
cu
CL
Nun
num
bers
ber of
/7 curvi
profile
s analy
'sed.
1
.218
/
\
/
\
/
\
^
226
1
\
\
/
5wi
7/7 /5/i7
'70''^
A
166
\
/
\
\
/
\
1
1
\
'7
1
y496
\
(
S
k
151
^
\
H
'ashinc,
/■/c/?, z;
..-^
\.
^472
247
y
^-v
\
'175
^o
1
ir banks, Alaska -—,
1 ^1
447
/
247^
^^^
247
M A
M
N D
Months of the Year
Figure 4.36. Frequency of occurrence of ground-based ducts.
by assuming a linear decrease of n within the duct. The coefficient c
is a constant and 7 is a function of the n gradient excess over the minimum
value of An/A/i = 1/a. Expressing X^ai in centimeters, and d in meters,
then
{Kl^ _ 0.157) 10;'
(4.25)
142
and
N CLIMATOLOGY
c = 2.514 X 102.
The maximum wavelengths trapped during ducting conditions were
estimated by (4.24) for the maximum duct thicknesses of figure 4.40.
These values, given in table 4.7, were determined for the month with the
maximum occurrence of ducts, thus allowing an estimate of the radio
frequencies likely to be affected by ducting conditions. Note, for ex-
ample, that the data of table 4.8 indicate that 1000 Mc/s rays (X = 30 cm)
will be trapped by 50 percent of the ducts regardless of location.
FAIRBANKS, ALASKA
Numbers on curves are the
total number of observed ducts.
\
,
{
\\
1
V
Max
18 \ \
\ j^Meon
1
iMin
FEB. MAY AUG NOV
WASHINGTON, D.C
t
\
r\
^\
'/
-21
V
Max
3
^
\
9
Mean
^
Min
FEB MAY AUG NOV
SWAN ISLAND
^^
Max
/
/
14
Mean
Z5,
30
^
4
/
Min
\
\
/
FEB MAY AUG NOV
Months of the Year
Figure 4.37. Angle of penetration of ground-based ducts.
GROUND-BASED ATMOSPHERIC DUCTS
143
< 260
I
1 1 1
FAIRBANKS, ALASKA
\ ' ' '
\ Numbers on curves are the
\ total r>umber of observed ducts
\
\
\
\
1
\
/
v
18 \
\\
\
^Max
^
^
\
^
\\Mean
1
1 1
WASHINGTON, D.C
1 1
i
J
\
/
\
/
\
/
\
/
\
/
\
/
.3 \
/ ^
\
1 /
\
7,
\
Mox
9
Mean
Min
1
SWAN ISLAND
1
Mox
^
^
/
1
1
> Mean
y
4
1
/Min
\
FEB MAY AUG NOV
FEB MAY AUG NOV FEB MAY AUG NOV
Months of the Year
Figure 4.38. Refractivity gradients of ground-based ducts.
Table 4.7. Estimated maximum wavelength trapped at do =
Station
Percentage of duets that trap wavelengths in centimeters equal to
or shorter than tabulated value
95
90
75
50
25
10
5
Fairbanks, Alaska (February).
Washington, D.C. (August)__.
Swan Island (August). . .
20.5
7
12
22.6
10
23
25.5
27
41.5
34.0
50
60
43.5
112
82.3
61.5
164
92
69.0
200
132
Table 4.8.
Ranges of
surface ducting gradients
Duct gradients
at Washington, D.C.
N units/km
February
N units/km
May
N units/km
August
TV units/km
November
Max
-200
-185
-175
-372
-230
-162
-395
-240
-158
—195
Mean
Min
-170
— 159
144
N CLIMATOLOGY
o
300
280
260
240
220
200
180
160
140
120
100
80
60
40
20
-Washington, D.C.
\
\
N
\,
N
\
"^
"--^
\
"^
^
^
^
^
\
\
N^
'^
^
-^
---Swan Island
X
^^
""^
«
F
'airb
ank.
?, Alaska —
>-v
- Washington
D.C
'
0! 0,2 0,5 I 2 5 10 20 30 40 50 60 70 80 90 95 98 99 995 99S999
Percentage of Observations That Equal or Exceed ttie Ordinate Value
Figure 4.39. Observed ground-based duct thickness.
The reader is cautioned that an atmospheric duct does not have the
sharp boundaries of a metallic waveguide. Thus the maximum wave-
lengths obtained by (4.24) do not represent cutoff frequencies but, as
Kerr is so careful to emphasize, merely suggest lower limits under the
assumptions of this rudimentary theory.
The results given above were derived from a consideration of radio-
sonde data. Although the radiosonde is not an extremely sensitive in-
strument, it is readily available and is the only source of climatic informa-
tion involving the temperature and humidity structure of the atmosphere.
It is believed that the radiosonde data will at least yield the climatic trend
of radio ducts as well as their probable temperature and humidity distri-
butions. Further, it is evident that the choice of stations will definitely
affect the percentage of ducts observed. For example, it is almost certain
that a greater percentage of ducts would be observed over water in the
subtropics than over Swan Island.
GROUND-BASED ATMOSPHERIC DUCTS
145
With these reservations in mind, the present study has shown:
(a) Ducts occur no more than 15 percent of the time.
(b) The annual cycle of the incidence of ducts is reversed for the arctic
and tropical stations studied. The arctic station has a wintertime maxi-
mum and the tropic station a summertime maximum. The temperate
station has a summertime maximum incidence of less than 5 percent.
(c) The maximum initial elevation angle is limited to 5.8 mrad with
a mean value of about 3 mrad.
(d) The steepest gradient of A^ observed is —420 A'^ units per kilometer.
(e) The maximum thickness of observed ducts is such as to trap radio
waves of 1000 Mc/s and above at all locations for at least 5 percent of the
observed ducts.
(f) Ducts in the arctic appear to be associated with temperature in-
versions at ground temperatures of —25 °C or less; temperate zone, with
the common radiation inversion and accompanying humidity lapse;
tropics, with a moderate temperature and humidity lapse for temperatures
of about 30 °C.
450
400
350
300
250
Q 200
E
•i 150
100
50
>
^
N
u-^
Vashin
gton
D.C
-.
\
N
s
X
\
\
\
^
^
^
\
^-^
'airba
nks,
Ale
iska
—Washington, D.C.
TS 1 1
^^Swan Island
^<>
0.1 0.2 0.5 I 2 5 10 20 30 40 50 60 70 80 90 95 98 99 995 99S 999
Percentage of Observations That Equal or Exceed the Ordinate Value
Figure 4.40. Maximum ground-based duct thickness.
146 A^ CLIMATOLOGY
4.5. A Study of Fading Regions Within the Horizon
Caused by a Surface Duct Below a Transmitter
4.5.1. Introduction
Among the factors influencing the choice of an antenna site for a micro-
wave receiver operating on a line-of-sight path is the desirability of
locating in a region relatively free from space-wave fadeouts. Although
line-of-sight propagation is normally characterized by steady, high, de-
pendable signals, deep, prolonged space-wave fadeouts are observed from
time to time. Since serious disruptions occur during such fadeouts, a
systematic discussion of fading phenomena is of considerable interest to
the radio circuit engineer.
There are [35] results in detail of a year's study of signal fadeouts
occurring at certain points within the horizon caused by ground-based
superrefractive layers. The path studied in this case [35] has its trans-
mitter at Cheyenne Mountain near Colorado Springs, Colo., at an eleva-
tion of 8800 ft. Receivers are at Kendrick, Colo., 49.3 mi from the
transmitter, and at an elevation of 5260 ft; Karval, Colo., 70.2 mi from
the transmitter, and at an elevation of 5260 ft; and Haswell, Colo., 96.6
mi from the transmitter, and at an elevation of 4315 ft. A study of the
profile of this path reveals that Kendrick is well within the radio horizon,
Karval is near, but still witnin the radio horizon, and Haswell is beyond
the radio horizon.
The Cheyenne Mountain study found that throughout the period of
observation, fadeouts (of 5 dB or more) occurred regularly at Karval on a
frequency of 1046 Mc/s, in conjunction with superrefractive A'^-profile
conditions, and coincided with enhanced field strengths at Haswell: A
particular instance of this, the night of June 21-22, 1952, is shown in
figures 4.53 and 4.54. Figure 4.53 shows the field strengths recorded
simultaneously in a 12-hr period at Haswell, Karval, and Kendrick. Here
it is to be noted that with the sudden intense rise in field strength at
Haswell, a progressively deepening fadeout appears in the Karval data
relative to the monthly median, while only insignificant changes are noted
at Kendrick. Meanwhile, figure 4.54 indicates the shift in the refrac-
tivity profile at Haswell (presumably the same for Kendrick and Karval)
towards conditions of superrefraction throughout the evening of 21-22
June, 1952.
The great number of fadeouts at Karval as compared to Kendrick
strongly suggested a dependence of fading on distance from the trans-
mitter within the horizon. The following sections describe the conditions
under which fading, variations in fading, and locations that favor the
occurrence of fading within the radio horizon (in the presence of a ground-
based duct) occur.
FADING REGIONS WITHIN THE HORIZON 147
Fading, or fadeout, of a radio wave may be defined as a drop in power
or field strength below a specified level of intensity. For a given site,
criteria may be set up by the communications engineer to establish the
magnitude of drop that defines the onset of fading conditions. This
magnitude usually ranges from about 5 to 30 dB below the specified level
of intensity. The specified level is derived from some ideal propagation
condition.
Fading is also a time-dependent occurrence. It may be classified as
(a) prolonged fading, which is fading of sufficient interval to cause con-
tinued communication disruptions, and (b) short-term Rayleigh fading,
which is only instantaneously observed by a receiver.
One of the main causes of deep fading of field strength within the
horizon as compared with the free space field strength value is the A^
structure of the atmosphere. Theoretically, the atmosphere can be con-
sidered horizontally homogeneous and in spherical stratifications con-
centric with the earth, and N can be considered to decrease exponentially
with increasing height above the earth's surface. However, in reality,
this picture of the atmosphere is rarely, if ever, realized because of the
synoptic meteorological conditions that are perpetually present. Strati-
fications caused by the synpotic meteorological pattern give rise to field
strength fading within the horizon by defocusing the lobe pattern of the
transmitter along a given path. Whenever the rate of change of refrac-
tivity from the surface value with height (called the gradient of A^ with
height) is less than —157 A^ units per kilometer, a "ducting" condition is
said to exist at the surface, and, as shown in figure 4.35, certain rays will
tend to be "trapped" or guided within the surface duct. It is this atmos-
pheric condition of surface ducting which will be further explored herein
with respect to fading within the radio horizon.
It should be realized that not all within-the-horizon fading has been
attributed to refractivity gradient discontinuities in the lower atmos-
phere. Misme [36] shows the influence of frontal effects and frontal
passage on signal fading within the horizon, the frontal passage even
occurring at a time when one would most expect a fadeout caused by a
surface duct.
4.5.2. Regions and Extent of Fading Within the Horizon
in the Presence of Superrefraction
Serious disruptions in reception from a transmitter above a duct to
a receiver within a duct can occur at particular points within the horizon.
It is these disruptions, and these locations, which are of interest to the
communications engineer establishing a given transmitting-receiving
path. Whenever these disrui)tioiis occur between a transmitter and
receiver within the horizon, a corresponding increase in the field strength
characterized by steady, high, dei)endable signals, is usually expected
148 N CLIMATOLOGY
beyond the horizon. The high signal strength is in keeping with the
properties of a surface duct. Nevertheless, deep, prolonged fadeouts
can occur in regions beyond the horizon as well as within the horizon.
Price [37] has theoretically determined regions of deep radio fading
associated with surface ducts, which he has termed "shadow zones." In
the case of a transmitter above a surface duct, a representation of what
occurs is shown in figure 4.43. It shows the location of a shadow zone
above the radio horizon line in the normal interference region. In the
interference region, a fadeout of signal strength due to the presence of
superrefraction must be compared with the value of the field when only
the interference pattern is present. Papers such as those by Norton [38]
and Kirby, Herbstreit, and Norton [39] give methods for calculation of
the normal interference field. Ikcgami [40] gives a more general method
for the calculation of received power in the presence of ground based ducts.
Ikegami's procedure is based on a simple geometrical optics ray-tracing
technique of determining the power relative to free space transmission
that is received at different locations from the transmitter. A more
refined, yet more complex, procedure is the field-strength calculations
along the lines followed by Doherty [41], in which a strict mode-theory
treatment of the problem reveals that geometrical optics is not sufficient
in the presence of refraction anomalies. Doherty expands techniques
originally considered by Airy [42] to determine relative field strengths in
the neighborhood of caustics (apparent ray intersections) that result
vicinal to refraction anomalies such as ducts.
In the procedure that follows, a model for the determination of the
location and extent of shadow zones, rather than the determinations of the
actual received field strengths and powers at any particular point, will be
given for conditions typical of the temperate climate to aid the radio cir-
cuit engineer in avoiding these troublesome interference areas.
Washington, D.C., is taken to represent an average temperate zone
climate. Fairly extensive work has been done in determination of various
ducting conditions at this station as well as with the average exponential
refractivity above this station. Therefore, Washington, D.C., will be
used as a model for all following calculations.
The gradient of A^ with height determines whether or not a surface
duct will exist. Therefore, a surface ducting atmosphere corresponding
to conditions at Washington, D.C., consisting of a ducting gradient up to
any desired height, and the N s = 313.0 exponential atmosphere [9] above
this height, have been chosen for this study. The ducting gradients
chosen are maximum, mean, and minimum values for February, May,
August, and November (representing the middle of winter, spring, sum-
mer, and fall, respectively) taken from figure 4.38 and given in table 4.8.
Also in this particular model the duct is assumed to be of uniform height
throughout the region considered in the calculations.
THEORY AND RESULTS
149
4.5.3. Theory and Results
In figure 4.41, although it is known that ray theory does not explain
what happens to the radio ray at the point A when the ray is tangent to
the top of the duct, because of the apparent ray intersection, it is sufficient
to define the first shadow zone as that zone between the two possible paths
of the ray tangent to the duct at a point A (this is the ray emitted from the
transmitter with an initial angle of ^o with the horizontal). This is possi-
ble because it can be readily seen that no radiation will be able to enter the
shadow zone, and thus a receiver located in the shadow zone will theoreti-
cally receive no signal from the transmitter. The portion of the "split"
ray that enters the duct at A will be reflected at point B to point C, where
it "splits" again. The distance, d from point A to C projected along the
earth's surface will be designated the "length" of the first shadow zone.
If the surface duct is homogeneous throughout, the ray will be symmetri-
cal about point B and, therefore,
d, = d. = d/2 = d,
(4.26)
where di and dz are the "half-lengths," dhi, of the shadow zone.
Figure 4.41. Shadow zone occurrence of a curved earth with the transmitter above
the duct.
150 A^ CLIMATOLOGY
Chapter 3 shows that the bending, ti,o, of the radio wave between any
two points in the atmosphere is given by
ri.2 = ^ + (01 - e,), (4.27)
where di,2 is the distance between the two points 1 and 2 (see fig. 4.42)
projected along the earth's surface, a is the radius of the earth, and 6i
and 62 are the angles in radians the ray makes with the horizontal at points
1 and 2 respectively. As described in chapter 3, Schulkin [43] obtained
Ti,2 in radians as
r,,, = ^^^ 7 ^^^ for < 00 < 10° (4.28)
where Arii = rii — 1, A/io = no — 1, d^ = {Si + 02)/2. In terms of refrac-
tivity, (4.28) becomes
{N, - N2) X 10-^ AN X 10-^ ., ^Q.
Tl.2 = a = a • (4.29)
Considering now the case of the half-length, dht, (4.27) becomes
ri.2 = ^ + e„ (4.30)
since 02 = at the top of the duct, and di = 6p. Now substituting (4.29)
in (4.30),
^^q^i^ = ^-' + 0„ (4.31)
since here dm = 0p/2. Rearranging (4.31)
dht = a
Since
'2AN X 10~'
_ dp (rad)
+ dp (rad) . (4.32)
dp = cos-^ ^^^ ^ J2[aN X 10-^ - ^'^ ~ ^h . (4.33)
(rad)
THEORY AND RESULTS
151
RADIO RAY
Figure 4.42. Geometry of radio ray refraction.
where ua is the index of refraction at the top of the duct, ta is the distance
from the center of the earth to the top of the duct, and no is the index of
refraction at the earth's surface, it is seen that dht is uniquely determined
from the surface gradient of A'^ with height inside the duct, provided the
height of the duct is known.
If one desires to know the distance, dh, for a particular height, h, from
the reflection point of the ray in the duct (point B in fig. 4.41), where dh
is such that
(4.34)
152
A^ CLIMATOLOGY
R^ 1 ST SHADOW ZONE
ZONE BEYOND THE HORIZON
DIRECT AND REFLECTED RAY
INTERFERENCE REGION
^GROUND BASED DUCT
FiGUEE 4.43. Regions of fading from different effects of the radio rays.
then (4.37) becomes
dh = a
2AN' X 10"
+ (e.
I
dp + 02
since 02 is no longer just equal to zero, but is such that
< 02 < 0p,
(4.35)
(4.36)
and AN' will no longer be the same as the AA'^ for the duct width. The
value of 02 may be determined from the formula:
2 2h
I
a
2AN'
(4.37)
In figures 4.44 to 4.46 the shadow zone half-lengths are given as a func-
tion of duct height for the various seasons of the year. As shown in
figure 4.39, the 50 percent level, or median of observed ground-based duct
thickness at Washington, D.C., is about 100 m. Figure 4.40 shows that
the percent level of observed maximum ground-based duct thickness is
about 200 for Washington, D.C. In table 4.9, the values oidht during the
four seasons are shown for the 100- and 200-m surface duct heights.
As previously mentioned, the atmosphere above the assumed duct was
taken to coincide with the N ^ = 313.0 exponential atmosphere [9] because
this value most nearly corresponds to the average annual N s value at
THEORY AND RESULTS
153
1,000
700
500
E
300
-^
200
z
X3
100
U
70
u
50
^
^
30
c/^
20
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" l"^-i==='^ [ ■ ■ ■ i .';■'■ !. i ' i 1 .; . 1
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1 11 . : j
-.<ik^. -\ ."--"^"^^^--11! 1 ■ ■
40
60 80 100 120 140
DUCT HEIGHT ,h, IN m
160
180 200
Figure 4.44. Shadow zone half-lengths for maximum ducting conditions at
Washington, D.C.
o
1,000
700
500
300
200
100
70
50
30
20
10
7
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:iEiiiii--|-ii:=iiiiiqEi==iiiiiEEi^|iiiii;.
20 40 60 80 100 120 140
DUCT HEIGHT . h , IN m
160
180 200
Figure 4.45. Shadow zone half-lengths for mean ducting conditions at
Washington, D.C.
154
N CLIMATOLOGY
Q
60 80 100 120 140
DUCT HEIGHT , h, IN m
180 200
Figure 4.46. Shadow zone half-lengths for minimum ducting conditions at
Washington, D.C.
Table 4.9. Half-lengths at Washington, D.C. for various duct heights and surface
gradient conditions
Cond.
Duct height
100 m
200 m
February
Max
Mean
Min _.
68.21
84.50
105. 35
96.46
119. 50
148. 98
May-
Max
Mean
Min
30.53
52.37
199. 42
43.17
74.05
281. 95
August
Max
Mean
Min
29.02
49.12
440. 73
41.04
69.46
622. 53
November
Max
Mean
Min
72.56
123. 93
313.61
102. 60
175. 24
443. 23
Washington, D.C. Figures 4.47 and 4.48 show the distance measured
along the earth's surface that a ray with an initial elevation angle of zero
will travel in the A^^^ = 313.0 exponential atmosphere. The value of
zero for an initial elevation angle is used because the ray can be thought
of as leaving the position A, tangent to the duct in figure 4.41 and
"arriving" at the transmitter, even though the reverse process is actually
what is occurring. Thus, from figures 4.47 and 4.48, knowing the height
of the transmitter, ht, and the height of the ground-based duct, Ka, the
THEORY AND RESULTS
155
400
1,000
2,000 3,000 4,000 5,000
HEIGHT ,h , IN m
6,000 7,000
Figure 4.47. Distance versus height above duct representing average Washington,
D.C., and Fairbanks, Alaska, atmosphere.
distance, do, that the radio ray travels from the transmitter to the top of
the duct is given by
d(ht -
hA):
(4.38)
i.e., the distance obtained by using the difference of ht and Ha as the value
of height on the abscissa in figures 4.47 and 4.48 gives the distance do.
The distance, dho, to the standard radio horizon from the transmitter is
dho = \^2ht
(4.39)
if ht, the transmitter height, is in feet and rf^o is in miles. Using this fact
and figures 4.47 and 4.48, table 4.10, which shows the ratio of do to dho,
Table 4.10. Ratio of onset point of fading zones from transmitter to distance of radio
horizon from the transmitter, do/dho, for various transmitter heights at Washington, D.C.
Height of
Duct height, meters
transmitter
10
20
50
100
200
200
300
500
800
1000
1500
2000
3000 -
5000
7000
0.795
.837
.878
.907
.918
.935
.944
.953
.959
.959
0.699
.759
.818
.860
.876
.900
.914
.928
.940
.943
0.512
.605
699
.765
.792
.831
.854
.880
.902
.911
0.300
.432
.565
.659
.697
.754
.787
.825
.859
.875
0.187
.375
.509
.563
.644
.692
.747
.799
.825
156
A^ CLIMATOLOGY
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40 80 120 160 200 240 280
HEIGHT, h, IN METERS
Figure 4.48. Distance versus height above dud representing average Washington, D.C.,
and Fairbanks, Alaska, atmosphere.
is derived for various transmitter heights and various duct heights. This
table is useful in determining the portion of the distance (measured from
the transmitter) within the horizon where fading due to the presence of
superrefraction has no effect on transmission.
SAMPLE COMPUTATIONS 157
4.5.4. Sample Computations
It can be seen from figure 4.52 that
hi < loss region < Ha + ho, (4.40)
where hi can be determined from figures like 4.49 to 4.51, which depend
upon the station location and the season of the year; hA is the height of
the duct; and /i2 is determined from figures 4.47 and 4.48. The following
example will illustrate the determination of the shadow zone.
As an illustrative problem, assume a 500-m transmitting antenna in the
spring of the year at 40°N latitude, and that one desires to place the re-
ceiving antenna at (a) 40 km and (b) 90 km from the transmitter. How
high should the receiving antenna be to be free from fading loss caused
by a ground-based superrefractive layer?
Since both of the distances required are within the standard horizon
distance, dho, which, for a 500-m transmitter is
dho = 92.181 km,
this is a case of within-the-horizon propagation.
Since the 40°N latitude position is within the temperate zone, use will
be made of the Washington, D.C., model ducting atmosphere. As pre-
viously mentioned, 100 m is the median mean duct thickness at Washing-
ton, D.C., and 200 m is the median maximum duct thickness observed at
Washington. These are the two duct thicknesses to use in this calcula-
tion. Also, since it is the spring of the year, figures 4.49 to 4.51 must be
used to determine hi.
For the 40-km distance, case (a), the subscript of (4.38) becomes
ht - hA = 500 - 100 = 400 m,
which is then entered on the abscissa of figure 4.47, where,
do = r/400 = 83 km.
For a 200-m-thick duct,
300 m.
158
N CLIMATOLOGY
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500
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'0 20 40 60 80 100 120 140 160 ISO 200
HEIGHT ,h , IN m
Figure 4.49. Height versus distance at Washington, D.C., for maximum midspring
(May) ducting conditions.
20
40
60 80 100 120 140
HEIGHT , h , IN m
160
180 200
Figure 4.50. Height versus distance at Washington, D.C., for mean midspring
(May) ducting conditions.
SAMPLE COMPUTATIONS
159
1,000
700
500
300
200
E
^ 100
^
70
-
'sn
■o
UJ
30
C)
■z.
k^U
<r
1-
co
10
Q
7
5
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IE
1
1
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'3
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!■^l r ■ \ i 1
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' : 1 : i 1. 1 : 1 i
ll^
55 m
"liuu mniuam
-
-
loc^^t I : :
K
J
-U-r-
^^^
y^.\ \ \ \ \]S-
=a=RTf'il200m|
-
=
qi:
^
~f^'
-^
--
-y'
^
^
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F^
^^^^pps: S:jr!lj^^_|,..j...j_.,...|....,._-__^
d
f ' '
<
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;
f\ '
y ^
rf--
(**>*
/
^
^
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/
y ^
If*
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-
1**^
^
j^
j^
^
is-"
1
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:.:
5
20
40
60 80 100 120 140
HEIGHT ,h , IN m
160
180 200
Figure 4.51. Height versus distance at Washington, D.C., for minimum midspring
{May) dxicting conditions.
Therefore, in this case,
do = dzQQ = 72 km.
In either case, as can be seen from figure 4.41, do is greater than 40 km.
Therefore, any height of the receiving antenna will be satisfactory at this
point, since the first shadow zone has not yet been reached.
For the 90-km distance, case (b), it can be seen that this distance is
beyond the onset point of the first shadow zone, and figures 4.49 to 4.51
must be used here.
Figures 4.49 to 4.51 are used differently depending upon which side of
the reflection point B of figure 4.52 one desires to locate an antenna. If
on the lefthand side of point B, one must subtract the desired distance
from the distance from B to the top of the duct traversed by the ray and
use this new distance to determine height, h, in figures 4.49 to 4.51. This
procedure must be followed every time the ray heads downward. If on
the righthand side of point B of figure 4.52 (or any other reflection point),
one simply uses the distance from the reflection point directly. One must
keep in mind that any previous half-lengths must be subtracted from the
desired distance from the transmitter, as well as subtracting do, in order
to obtain the distance used in figures 4.49 to 4.51.
160
N CLIMATOLOGY
'0 I ^h£ n
Figure 4.52. Geometry of the shadow zone loss region.
For the maximum midspring ducting conditions (fig. 4.49) and the
100-m-thick duct, recalling do = 83 km, the antenna is to be located
90
83 = 7 km
beyond the onset point of the shadow zones. For the 200-m duct, the
antenna is located
90 - 72 = 18 km
beyond point A of figure 4.52. From figure 4.44, the half-length traveled
by a ray in a 100-m duct under maximum midspring ducting conditions
is 30.5 km. Thus, as described above, the distance used in figure 4.49
to find h is
30.5 - 7.0 = 23.5 km
because at 25 km the ray is still sloping dov/nward and hence is on the
lefthand side of point B of figure 4.52. From figure 4.44, dht = 43 km
for a 200-m duct; therefore, the distance used in figure 4.49 is
43
18 = 25 km.
Similarly, utilizing figures 4.45 and 4.46 for mean ducting and minimum
ducting conditions for a 100-m and 200-m duct:
Figure 4.50 distance to be used for the 100-m duct = 45 km,
Figure 4.50 distance to be used for the 200-m duct = 56 km,
Figure 4.51 distance to be used for the 100-m duct ^ 191 km.
Figure 4.51 distance to be used for the 200-m duct = 262 km.
SAMPLE COMPUTATIONS
161
Figure 4.53. 1 ,046-Mc/s fields for the night of June 21-22, 1952, at Kendrick, Karval,
and Haswell, Colo.
and hence
hi for a 100-m duct under maximum spring ducting conditions = 92.3 m,
hi for a 200-m duct under maximum spring ducting conditions = 166.0 m,
hi for a 100-m duct under mean spring ducting conditions = 96.0 m,
hi for a 200-m duct under mean spring ducting conditions = 187.0 m,
hi for a 100-m duct under minimum spring ducting conditions = 99.0 m,
and
hi for a 200-m duct under minimum spring ducting conditions = 199.0 m.
Figures 4.47 and 4.48 will be used to determine /i2 from the total
distance, d. The distance to be used is obtained by subtracting the dis-
tance, do, from the desired distance. For the 100-m duct, do = 83 km and
d = 90 - 83 = 7 km.
Likewise for the 200-m duct, do = 72 km and
rf = 90 - 72 = 18 km.
Thus it is found that for the model assumed:
(a) /i2 for the 100-m duct height case will always be 3 m,
(b) /i2 for the 200-m duct height case will always be 18 m.
Since from (4.40)
hi < loss region < /i^ + /12.
162
N CLIMATOLOGY
300
280
260
240
220
200
180
160
< 140
120
100
tTS tao tat IM tn >oo >ae
ITS tlO tM »0
Refractive Index in Units of N = (n-l)xlO®
Figure 4.54. Vertical distributions of refradivity at Haswell, Colo., for June 21-22, 1962
Therefore, in answer to part (b) of the problem, for maximum ducting
conditions with a 100-m-thick duct:
92.3 m < loss region < 3 + 100 m, or
92.3 m < loss region < 103 m.
INTRODUCTION 163
With a 200-m-thick duct :
166 m < loss region < 18 + 200 m, or
166 m < loss region < 218 m.
For mean ducting conditions with a 100-m-thick duct:
96 m < loss region < 103 m.
With a 200-m-thick duct:
187 m < loss region < 218 m.
For minimum ducting conditions with a 100-m-thick duct:
99 m < loss region < 103 m,
with a 200-m-thick duct :
199.3 m < loss region < 218 m.
4.6. Air Mass Refractive Properties
4.6.1. Introduction
The foregoing material on refractive index climatology concerned the
geographical variation of mean A'' (or A^'o) over the surface of the earth
and the three-dimensional distribution of mean A'^ over North America.
This section develops another aspect of the climatological variation of the
refractive index: the mean profile of A^ in various air masses. The
meteorologist defines air masses as bodies of air in the troposphere having
approximately homogeneous character at the surface. An air mass is de-
scribed in terms of its origin, as, for example, polar continental or tropical
maritime.
Profiles of A'^ and values of bending of radio rays are given in terms of
departures from normal for various air masses.
Recent studies have led to the evaluation of radar elevation angle
errors in different climates and air masses [43, 44]. In general, these
studies did not emphasize the relation between air mass refractive index
structure and the refraction of radio waves within the air mass. For
example, mean angular bendings for radio rays passing completely through
164
N CLIMATOLOGY
20 40 60 80 100 120 140 160 180 200
ANGLE OF ARRIVAL OR DEPARTURE, ^o.lmrad)
Figure 4.55. Comparison of total angular bending of radio rays within air masses.
After Schulkin [1952].
the earth's atmosphere are given for extremes of air mass types and a
range of initial elevation angles, do, on figure 4.55 [43]. It is seen that
maritime tropical air produces 30 percent more bending than continental
polar air at initial angles or arrival or departures of about 10° (175 mrad)
and that this difference increases to about 70 percent at zero initial
elevation angle. Further, the magnitude of the total bending increases
rapidly with decreasing values of ^o-
Figure 4.55 does not make it clear that differences in radio ray refrac-
tion arise from differences in the refractive index profiles of the two air
masses. It is the purpose of this section to show that various air masses
have characteristic refractive index profiles and that the radio ray refrac-
tion within each air mass is mirrored by the difference of the actual re-
fractive index structure from a standard atmosphere of exponential form.
4.6.2. Refraction of Radio Rays
The angular bending of a radio ray, ti,2 (see fig. 4.56) between two
points in the atmosphere of refractive indices rii and n2 is given by the
REFRACTION OF RADIO RAYS
165
RADIO RAY
Figure 4.56. Geometry of radio ray refraction.
geometrical optics formula [45, 43],
Ti,2 = — f cot 9 dn
(4.41)
under the assumption that dn/n = dn and 6 is the local elevation angle.
It is also assumed that n is spherically stratified and concentric with the
earth. The value of cot 6 is determined from Snell's law, and ti,2 is com-
pletely determined from a knowledge of the variation of n with height.
An empirical formula to describe the long-term average variation of n
166 A^ CLIMATOLOGY
with height is
WW) = RA) - 1] 10« = 313 exp {-V6.95) (4.42)
where h is the height above the earth's surface in kilometers and 313 is the
long-term average value of (n — 1) 10^ at the earth's surface for the United
States. In practice it is convenient to utilize this average N{h) function
to refer the observed N{h) data to the common level oi h = by means of
A{h, 313) = N{h) + 313 [1 - exp {-h/Q.95}]. (4.43)
Chapter 1 shows that A{h,31S) is analogous in concept to potential tem-
perature but utilizes the normal A'^ gradient rather than the adiabatic
gradient of the potential refractive modulus of Lukes [46] and Katz [20].
The notation A{h, 313) is used to indicate that A is determined from the
refractive index at the height h and the 313 exponential atmosphere. It
has been found to be advantageous to use several atmospheres of exponen-
tial form in applications involving different climatic regions [47]. The
particular form of (4.43) emphasizes departures of A^ structure from
normal as shown by recent studies of synoptic variations of N(h) about
frontal systems [21] and, in addition, permits direct calculation of radio
ray bending for any observed refractive index structure. When A{h 313)
is introduced into (4.41) for ray bending, one obtains (see chapter 3,
sec. 3.10).
r,, = - L ' IQ "^^^^ dAih, 313) + Tih, 313) (4.44)
JA^ n
which indicates that the bending can be regarded as the resultant of the
bending in the normal atmosphere, t(/i, 313), and a perturbation compo-
nent representing departures of refractive index structure from that of the
313 exponential atmosphere. The value of r(/i, 313) can be obtained from
refraction tables [9], and the perturbation term can be evaluated by
graphical methods to yield an overall accuracy of a few percent in esti-
mating ri,2 [47].
Some A(h, 313) profiles were prepared from typical temperature, pres-
sure, and humidity distributions within a range of air masses as published
in the literature [48, 49, 50] and climatic summaries of upper air data [31].
Two of these profiles, one for maritime tropical air and the other repre-
senting continental tropical air, are plotted on figures 4.57 and 4.58 to
represent the extremes of A{h, 313) profiles. These profiles clearly show
that the two air masses have quite different refractive index structures.
The difference is most pronounced near the ground. At heights of 20
km, however, A{h,313) rapidly approaches the asymptotic value of 313,
REFRACTION OF RADIO RAYS
167
40
36
32
28
E 24
- 20
D
^ 16
12
81-
4 -
A(h,3l3)
SAN JUAN, W.I.
(July Average)
'A(h,3l3)
= 3/3
''313-^
Bn=0
310 320 330 340 350 360 370 380 390
A(h, 313), N UNITS
I \ \ L \ \ \ I
-8
■6 -5 -4 -3 -2 -I
T313-T (mrad)
Figure 4.57. Comparison of A(h,313) profiles with departures from normal hendirig
of radio waves for maritime tropical air.
After Byers [1944].
168
N CLIMATOLOGY
40
36
32
28
EL PASO, TEXAS
I (June Average)
-A (h, 3/3 J
y
/
':^
310 320 330 340 350 360 370 380
A(h,3l3),N UNITS
-2
12 3 4
T3|3-T(mrad)
Figure 4.68. Comparison of A(h,313) profiles with departures from normal bending
of radio waves for continental tropical air.
After Ratner (1945).
REFRACTION OF RADIO RAYS
169
regardless of air mass type. The radio ray bending is also plotted on
these figures relative to the value expected in the Nih) = 313 exp
{ — /i/6.95} atmosphere. Near the ground, bending departures are seen
to be mirrored by the A unit profiles. That is, a negative gradient of
A'^ or yl produces a positive increase of bending. Above a few kilometers,
however, the bending departures approach a fixed, usually nonzero, value.
It is apparent that the asymptotic value of T313 — r at large heights is
determined by the bending in the first few kilometers, where 60 to 75
percent of the low-angle bending normally occurs. The marked differ-
ences in air mass n structure and bending are confined, therefore, to the
first few kilometers above the earth, as is illustrated by figure 3.16. The
distribution of the meteorological elements within each air mass is re-
flected in the A{h, 313) profiles. For example, the steep humidity
gradient characteristic of maritime tropical air is reflected by the rapid
decrease of A{h, 313) with height. Comparatively, the high ground-level
temperatures and rapid decrease of temperature with height in continental
tropical air are reflected by the increase of A{h, 313) with height wdthin
that air mass. These, and .4-profile characteristics of other air masses,
are listed in table 4.11 and are evident from the form of the equation for
the refractive index at radio frecjuencies [see chapter 1, (1.20)].
Table 4.11. Refractive characteristics of air 7nasses.
Ref-
erence
Superior (S) S/mT, [48]
typical of Oulf coast
(Lake Charles, La.).
Meteorological characteristics
Formed from subsidence of high-
level air with resulting dry adiabatic
temperature lapse rate and low hu-
midity. Often found overlying other,
more humid air masses that show a re-
sultant characteristic drying with
height in the lower levels.
Refractive characteristics
Rapid decrease of N and A
with height in the lowermost
layer produces a superrefrac-
tive* layer. The overlying
superior air is nearly normal in
bending characteristics.
Continental Polar (cP)
source region.
Characterized by low temperature
and humidity with a pronounced tem-
perature inversion at the surface
created by progressive nocturnal cool-
ing during the arctic night.
Ground-based superrefrac-
tive layer arising from the tem-
perature inversion.
Maritome Polar (mP), [48]
typical low-level
ground modification
(Seattle, Wash.).
Cool air, nearly saturated to a height
of several kilometers. Example shows
typical drying in lower levels when
this air mass moves over land. Over-
lying mP has increasing humidity and
decreasing temperature with height.
Continental Tropical
(cT) source region
(El Paso, Tex.).
[51]
Characterized by superheated lower
layers, rapid decrease of temperature
with height, and very low humidity.
Surface layer produces near-
normal bending. Increasing
N in the overlying mP air pro-
duces an elevated subrefrac-
tive layer.
Strong temperature lapse
produces a subrefractive layer
reaching to several kilometers.
Maritime Tropical
(mT) source region
(Pensacola, Fla.).
[48]
Relatively warm air, high water-
vapor content in lowest layer, which
decreases rapidly with height. Mod-
erate changes in temperature and hu-
midity structure produce large re-
fractive gradient changes.
Strong superrefraction to
great heights arising chiefly
from rapid decrease of hu-
midity with height.
'Normal refraction is taken to mean t(/i, 313), superrefraction to mean j-(ft)>7(h, 313), and subrefraction
t(/i)< r(ft, 313).
170 N CLIMATOLOGY
The average decrease of pressure, temperature, and humidity with
height produces a normal decrease of A'^. If, however, the temperature
increases with height, as in a temperature inversion, A^^ decreases more
rapidly and radio rays are superrefracted, dependent, of course, upon the
vapor pressure. Conversely, an unusually rapid decrease of temperature,
or an mcrease of humidity, with height produces a subnormal decrease of
A^ with height, or subrefraction of radio waves. In any event it is evident
from these figures that x4-profile effects on ray bending are most pro-
nounced at do = 0, are significantly less pronounced at do = 52 mrad
('~3°), and continue to diminish with increasing do until, at ^o = 7r/2,
there is no bending at all and consequently all departures are zero.
4.6.3. Conclusions
The work of Schulkin and others has shown that characteristic total
bending differences in radio ray refraction exist between various air
masses. The present study extends Schulkin's conclusion by identifying
abnormal bending of radio rays with departures of refractive index struc-
ture from average in the lowermost layers of the air masses. Considera-
tion of departures of both ray bending and refractive index structure from
their value in a standard exponential atmosphere results in a suitable
method of cataloging air masses in terms of either refractive index struc-
ture or bending characteristics.
4.7. References
Resolution No. 71 (1948), Conf. of Directors, Internal. Meteorol. Organization
(Lausanne, Switzerland).
World Atlas of Sea Surface Temperature Charts (1944), Hydrographic Office
Publ. 225, chap. 4, No. 2.
Atlas of climatic charts of the oceans (1938), U.S. Weather Bureau Publ. 1247,
Washington, D.C.
Johnson, W. E. (Nov.-Dec. 1953), An analogue computer for the solution of the
radio refractive index equation, J. Res. NBS 51, No. 6, 335-342.
Gerson, N. C. (1948), Variations in the index of refraction of the atmosphere,
Geofis. Pura Appl. 13, 3-4.
Bean, B. R. (Apr. 1953), The geographical and height distribution of the gradient
of refractive index, Proc. IRE 41, No. 4, 549-550.
Misme, P. (Nov.-Dec. 1958), Essai de radioclimatologie d'altitude dans le nord
de la France, Anna. Telecommun. 13, No. 11-12, 303-310.
Smithsonian Meteorological Tables (1951), Table 63, Sixth Revised Ed., Book
(Washington, D.C).
Bean, B. R., and G. D. Thayer (1959), CRPL Exponential Reference Atmosphere,
NBS Mono. 4.
Pickard, G. W., and H. T. Stetson (1959), Comparison of tropospheric reception,
J. Atmos. Terrest. Phys. 1, 32-36.
Pickard, G. W., and H. T. Stetson (1950), Comparison of tropospheric reception
at 44.1 Mc with 92.1 Mc over the 167-mile path of Alpine, New Jersey to
Needham, Mass., Proc. IRE 38, No. 12, 1450.
REFERENCES 171
Bean, B. R. (1956), Some meteorological effects on scattered radio waves, IRE
Trans. Commun. Syst. CS4, 32-38.
Onoe, M., M. Hirai, and S. Niwa (Apr. 1958), Results of experiment of long-
distance overland propagation of ultra-short waves, J. Radio Res. Labs. 5, 79.
Gray, R. E. (Jan., Feb., Mar. 1959), The refractive index of the atmosphere as a
factor in tropospheric propagation far beyond the horizon, IRE Natl. Conv.
Record, Pt. 1, 3-11 (1957), Elec. Commun. 36, No. 1, 60.
Norton, K. A. (Mar. 1956), Point-to-point radio relaying via the scatter mode of
tropospheric propagation, IRE Trans. Commun. Syst. CS-4, 39-49.
Schelleng, J. C, C. R. Burrows, and E. B. Ferrell (Mar. 1933), Ultra-short-wave
propagation, Proc. IRE 21, 427-463.
Anderson, L. J. (1958), Tropospheric bending of radio waves. Trans. Am. Geo-
phys. Union 39, 208-212.
Snedecor, G. W. (1946), Statistical Methods, Book, 4th Ed., pp. 97-98 (Iowa
State College Press, Ames, Iowa).
Bean, B. R., and L. P. Riggs (July-Aug. 1959), Synoptic variation of the radio
refractive index, J. Res. NBS 63D (Radio Prop.), No. 1, 91-97.
Katz, I. (1951), Gradient of refractive modulus in homogeneous air, potential
modulus, Propagation of Short Radio Waves by D. E. Kerr, Book, pp. 198-199
(McGraw-Hill Book Co., Inc., New York, N.Y.).
Bean, B. R., L. P. Riggs, and J. D. Horn (Sept.-Oct. 1959), Synoptic study of the
vertical distribution of the radio refractive index, J. Res. NBS 63D (Radio
Prop.), No. 2, 249-258.
Jehn, K. H. (June 1960), The use of potential refractive index in synoptic-scale
radio meteorology, J. Meteorol. 17, 264.
Bean, B. R., and J. D. Horn (Nov.-Dec. 1959), The radio refractive index near
the ground, J. Res. NBS 63D (Radio Prop.), No. 3, 259-273.
duCastel and P. Misme (Nov. 1957), Elements de radio cUmatologie, L'Onde
Electrique 37, 1049-1052.
Hay, D. R. (Dec. 1958), Air mass refractivity in central Canada, J. Phys. 36,
1678-1683.
Tao, K., and K. Hirao (Mar. 1960), Vertical distribution of radio refractive index
in the medium height of atmosphere, J. Radio Res. Labs. 7, No. 30, 85-93.
Fehlhaber, L., and J. F. Grosskopf (1959), Beitrage zur Radioklimatologie
Westdeutschlands, technische Bericht 5546, FTZ.
Bean, B. R., and R. M. Gallet (Oct. 1959), Applications of the molecular re-
fractivity in radio meteorology, J. Geophys. Res. 64, No. 10, 1439-1444.
Bean, B. R., J. D. Horn, and A. M. Ozanich, Jr. (1960), Climatic charts and data
of the radio refractive index for the United States and the world, NBS Mono. 22.
Report 147 (1959), IX Plenary Assembly, International Radio Consultative
Committee, pp. 299-337 (Los Angeles, Calif.).
Booker, H. G., and W. Walkinshaw (1947), The mode theory of tropospheric
refraction and its relation to wave guides and diffraction. Book, Meteorological
Factors in Radio Wave Propagation, pp. 80-127 (The Physical Society and
Roy. Meteorol. Soc, London, England).
Bremmer, H. (1949), Terrestrial Radio Waves, Book, pp. 131-138 (Elsevier
Publ. Co., New York, N.Y.).
Cowan, L. W. (1953), A radio climatological survey of the U.S., Proc. Conf. Radio
Meteorol. (Univ. of Texas, Austin, Tex.).
Kerr, D. E. (1951), Propagation of Short Radio Waves, Book, pp. 9-22 (McGraw-
Hill Book Co., Inc., New York, N.Y.).
Bean, B. R. (May 1954), Prolonged space wave fadeouts at 1046 Mc observed in
Cheyenne Mountain propagation program, Proc. IRE 42, 848-853.
172 N CLIMATOLOGY
[36
[37
[38
[39
[40
[41
[42
[43
[44
[45
[46
[47
[48
[49
[50
[51
Misme, P. (Apr. 1956), Methode de mesure thermodynamique de I'indice de
refraction de I'air-description de la radiosonde MDI, Ann. Telecommun. 11,
No. 4, 81-84.
Price, W. L. (July 1948), Radio shadow effects produced in the atmosphere by
inversions, Proc. Phys. Soc. 61, 59-73.
Norton, K. A. (Dec. 1941), The calculation of ground-wave field intensity over a
finitely conducting spherical earth, Proc. IRE 29, 623-639.
Kirby, R. S., J. W. Herbstreit, and K. A. Norton (May 1952), Service range for
air-to-ground and air-to-air communications at frequencies above 50 Mc,
Proc. IRE 40, No. 5, 525-536.
Ikegami, F. (Aug. 1959), Influence of an atmospheric duct on microwave fading,
IRE Trans. Ant. Prop. AP-7, 252-257.
Doherty, L. K. (1952), Geometrical optics and the field at a caustic with applica-
tions to radio wave propagation between aircraft, Res. Rept. EE138 (Cornell
Univ., Ithaca, N.Y.).
Airy, G. B. (1938), On the intensity of light in the neighbourhood of a caustic,
Cambridge Phil. Trans. 6, 379-402.
Schulkin, M. (May 1952), Average radio-ray refraction in the lower atmosphere,
Proc. IRE 40, No. 5, 554-561.
Fannin, B. M., and K. H. Jehn (May 1957), A studj' of radar elevation angle
error due to atmospheric refraction, IRE Trans. Ant. Prop. AP-5, 71-77.
Smart, W. M. (1931), Spherical Astronomy, ch. 3, Book (Cambridge Univ. Press,
London, England).
Lukes, G. D. (1944), Radio meteorological forecasting by means of the thermo-
dynamics of the modified refractive index. Third Conf. on Prop., NRDC,
pp. 107-113 (Committee on Propagation, Washington, D.C.).
Bean, B. R., and E. J. Button (May- June 1960), On the calculation of departures
of radio wave bending from normal, J. Res. NBS 64D (Radio Prop.), No. 3,
259-263.
Byers, H. R. (1944), General Meteorology, Book, pp. 255-277 (McGraw-Hill
Book Co., Inc., New York, N.Y^).
Willett, H. C. (1944), Descriptive Meteorology, Book, pp. 190-220 (Academic
Press, Inc., New York, N.Y.).
Trewartha, G. T. (1943), An Introduction to Weather and Climate, Book, pp.
206-215 (McGraw-Hill Book Co., Inc., New York, N.Y.).
Ratner, B. (1945), Upper air average values of temperature, pressure, and relative
humidity over the United States and Alaska (U.S. Weather Bureau).
Chapter 5. Synoptic Radio Meteorology
5.1. Introduction
This chapter treats the variation of refractive index structure in the
troposphere with synoptic tropospheric disturbances. Within the scope
of the synoptic field are timewise and spacewise variations in the atmos-
phere from microscale fluctuations to broad-scale systems of weather map
dimensions.
The microscale fluctuations of the refractive index are those that one
would expect to observe at a particular point along a radio path that
reflect detailed terrain and weather conditions in the immediate vicinity
of, say, the transmitter or receiver site.
Mesoscale variations, by way of contrast, are those which cover tens
of kilometers and thus encompass a substantial portion of a radio path.
Examples of this type of variation are land-sea breeze effects and con-
vection cells of thunderstorm activity.
Large-scale weather systems, affecting vast areas, perhaps even on a
continental scale, fall under the classification of macroscale variation.
Examples of this type of activity in the atmosphere are sweeping air mass
changes and frontal systems traversing thousands of kilometers on the
earth's surface. A detailed analysis of such a system is given as an illus-
tration later in this chapter.
5.2. Background
The problem of determining the vertical and horizontal distribution of
the radio refractive index has engaged the attention of radio meteorologists
on an international scale for the better part of two decades [1, 2, 3, 4, 5,
By analysis of current synoptic conditions from standard weather
charts, one may ascertain the air mass type appearing over a given region
and, likewise, may predict with reasonable accuracy the air mass type
expected over a particular locale in, say, 24 hours. Then, from the air
mass profile characteristics table of section 4.4, one is able to estimate
what the departures of refractive index (and radio ray bending) from
normal will be over a certain region. The bending predictions permit an
Figures in brackets indicate the literature references on p. 224.
173
174 SYNOPTIC RADIO METEOROLOGY
estimate of refraction errors and the introduction of appropriate correc-
tions for radio range and elevation angle errors for radio navigational
equipment.
VHF-UHF radio field strengths beyond the normal radio horizon will
also differ from air mass to air mass. It has been known for many years
that the seasonal cycle of VHF radio field strengths received far beyond
the normal radio horizon were correlated with the refractive index [7, 8,
9, 10] and that significant changes in field strength level are observed from
air mass to air mass [11, 12, 13.] Speaking about signal level on a 60
Mc/s beyond the horizon radio path near Boston, Mass., Hull [11] states
that during the winter low signal levels prevail during the presence over
the path of fresh polar air. Periods of high signal occur when a cold, dry
polar air mass is overrun by warm, moist air of tropical maritime origin.
Hull's analysis represents early recognition of refraction and reflection
phenomena on a synoptic scale. Later work on seasonal changes of fields
and A'^ represents, in a way, a summary of synoptic conditions over a
period of time.
Gerson [2] was one of the first to consider the variation of the radio
refractive index, n, in terms of seasonal and air mass changes. Gerson
divided n into two parts, one density-sensitive and the other moisture-
sensitive. This division is equivalent to the wet- and dry-term separation
of A'^ = (n — 1) 10^ in chapter 4, section 1. Gerson was able to measure
seasonal thermal changes by variation in the dry term and seasonal
moisture changes by observed variation in the wet term. Gerson pre-
pared graphs showing a sinusoidal variation of the dry term with a warm
season trough and a cool season crest, indicating density changes in inverse
proportion to the temperature. The wet-term component of n, on the
other hand, was observed to attain its maximum during the warm season
when the dry term was at its minimum. In arctic and antarctic locales,
the surface variation of the wet term was found to be quite small while in
temperate and tropical climates there was a sizable annual variation of
the moisture component. As a pertinent aside, Yerg [14] showed that
even during the long, cold, arctic night, vertical variations in moisture
made significant contributions to the A'' profile. Apparent ducting
gradients, obtained by neglecting the wet term at low temperatures may,
in actuality, be only slightly more refractive than standard.
Continuing his investigation, Gerson next turned to the analysis of
refractive index changes within various air masses. Using air mass data
available in the meteorological literature of the day, he charted mean
refractive index profiles for different air mass types. The largest initial
values and also the largest vertical gradients of n occurred with tropical
maritime air. The air mass with the weakest gradients and, therefore,
the poorest refraction properties was found to be the polar continental
type. Common to all of Gerson's air mass refractive index graphs is an
BACKGROUND 175
approximately exponential decrease of n with respect to height. Schel-
leng, Burrows, and Ferrell [15] attempted to remove this systematic n
decrease with height by utilizing a correction factor of linear form,
although, as has been seen in chapter 3, this leads to serious overestima-
tion of refraction effects in the more modern problem of satellite tele-
communications.
The work of Hay [6] confirmed and extended the observations Gerson
had made concerning air mass profiles. Measuring air mass character-
istics at Maniwaki, Quebec, Hay concluded that each large-scale air mass
type in central Canada has a distinctive refractive index profile. Hay
determined the height distribution of A'' for four basic air masses by fitting
a second-degree polynominal to each of the four sets of air mass data,
indicating that an average A'' profile cannot be approximated effectively
by a linear curve except over small height increments. Hay obtained
further discrimination by constructing a "dry-term" curve for each air
mass. The dry-term curves display a smaller standard deviation than
the total A'^ curves for all air masses except the continental arctic which
is, of course, itself very dry. The largest variations in total A'' are due
to fluctuations in the wet term. The saturation vapor pressure is approx-
imately an exponential function of the temperature, so that during the
warm season normal temperature changes cause the saturation vapor
pressure and, therefore, the wet term is vary sharply.
Contained also in Hay's paper [6] is a discussion of the use of A^ pro-
files to estimate corrections for radio ray refraction by use of a table of
mean effective earth's radius factors for each of four air masses within
95 percent probability limits.
Misme [5, 16] has had an interest in synoptic radio meteorology in con-
nection with telecommunications networks in France and North Africa.
Interests of this French radio engineer include vertical gradients of the
refractive index and atmospheric reflection of radio waves.
A reasonable question at this point is, to what extent are model ex-
ponential atmospheres applicable to various climatic zones around the
earth? While it is readily apparent that individual profiles vary widely
from any sort of exponential norm, there is an increasing backlog of
experimental evidence [17] that shows that long-term averages of synoptic
refractive index variations do tend toward an exponential form with
respect to height. Indeed, the 5-year mean A'^ profiles of figures 5.1 and
5.2, representing arctic and tropical climates respectively, bear out this
contention by showing close agreement with models developed from mid-
latitudinal data. Indeed, these examples verify the results obtained by
Gerson [2] for various air masses.
Randall [4] investigated the relationship of surface meteorological data
to surface A'^, N s, and field strength in the FM frequency band. The
results described were drawn from a very limited data sample that covered
176
SYNOPTIC RADIO METEOROLOGY
400
300
N 200
100 -
78°-50'N., 103°- 50' W.
i
FEBRUARY 1953-1957
w
v\
v\
234 PROFILES, Ns=332.5 _
.^N(h)
—
^^ ^UPPER AND LOWER
—
VV STANDARD DEVIATION
^^SU-IMITS
EXPONENTIAL REFERENCE^ ^^^^^
ATMOSPHERE, N^ =332.5
2 4 6 8 10 12 14
h(knn)
Figure 5.1. Mean N distribution for Isachson, N.W.T., Canada.
less than a month during the summer of 1947. Within the framework of
this speciahzed study, however, Randall found that polar continental air
masses were associated with low field strengths and low A^^ while tropical
maritime air masses were associated with high field strengths and high
N s, as shown on figure 5.3. Randall advanced the hypothesis that the
observed field strength changes were due to the existence of characteristic
A^ profiles typical of each air mass type. Randall was curious also as to
the behavior of A'' and radio fields during the passage of fronts and squall
lines. Figure 5.4 shows the results of this investigation and indicates that
BACKGROUND
177
definite field strength changes may occur during frontal and squall line
passage. Caution should again follow in the interpretation of these
results since they represent but a single example of frontal and squall line
passage.
Gray [18] made measurements of correlation of A^^ and the gradient on
N with path losses at UHF frequencies for observing points representative
of various climatic areas around the world and concluded that changes in
transhorizon telecommunications are strongly dependent on atmospheric
changes.
400
300
2''-46'S., I7r-43'W
FEBRUARY 1953-1957
274 PROFILES, Ns=37l.3
N 200
100
'UPPE^ AND LOWER
STANDARD DEVIATION
LIMITS
EXPONENTIAL REEERENCE-
ATMOSPHERE, Ns=37l3
2 4 6 8 10 12 14
h (km)
Figure 5.2. Mean N distribution for Canton Island, Pacific Ocean.
178
SYNOPTIC RADIO METEOROLOGY
;| i
is M
It
is
15
13
09
0.5
PLOTTING KEY
+
Averoge wmd speed equal
to or more thon lOmph
cP
mP
Polar conllnen'ol oir mass
Polar maritime air mass
mT
Tropicol morifime air moss
mT
+
+ mP
ml
♦mT
^^B
—
cPamP
cPar
nP ''
cPsmT^ .
nT
+■
mT
cP
cP
_^
^
cPamT
cPamT
1
mT
mP
^^--^^^
cPamT
rnT mT
+ +
*mT
A-^
--
1- + f
cPcPcP
+
cPamT
1-
cP
f
cPSmr
1
1
320
330
340
580
39C
Hour// Mean Surface Refracfivity
Figure 5.3. Field strength versus refractivity.
Scatter diagram of select hourly median field strength vs. hourly mean refractivity July 17 to Aug. 8, 1947.
In another article Gray [19] considered radio propagation and related
meteorological conditions over the Caribbean Sea. Utilizing the effective
earth's radius factor as a representative index, Gray presented an em-
pirical curve of annual median scatter loss versus effective propagation
distance designed to fit both Caribbean and temperate regions. Effective
distance as defined by Gray is the angular distance in radians multiplied
by the radius of the earth modified for normal refraction. Gray reports
that the refractive gradient in the first 100 m is in general the determining
factor in median scatter loss for transhorizon telecommunications, as one
would expect from earlier refraction studies [20].
Other studies on refraction problems during the 1950's have led to
systematic computation of refraction effects and to significant applica-
tions such as the evaluation of radar elevation angle errors in differing air
masses and climates [21 , 23]. Schulkin [21] advanced a practical and very
fundamental method for numerical calculation of atmospheric refraction
(radio ray bending) from radiosonde data. Figure 4.55 [after Schulkin]
gives mean angular bendings for radio rays passing completely through
the earth's atmosphere for two extremes of air mass type. Fannin and
Jehn concluded that a particular refractive index profile depends on air
BACKGROUND
179
mass type and the climatic controls of season and latitude, as is substan-
tiated by mean A^ profiles for 34 weather observing sites located in or near
four distinct air mass source regions about the world. These data show
that a definite difference does exist in the profiles of various air masses.
Refraction effects were found to be largest in tropical maritime air, inter-
mediate in polar continental and polar maritime, and least in tropical
continental. Fannin and Jehn also published graphs showing day-to-day
variations in profiles representing the effects of air mass changes over a
given observing station and graphs of diurnal profile variations.
In other studies along these lines, Bean, Horn, and Riggs [23], demon-
strated that radio ray refraction within the lower layers of an air mass is
mirrored by the difference between the observed refractive index structure
and that of a standard atmosphere. Figure 5.5 shows a graph of bending,
T, plotted together with a modified refractive index profile for an example
of summertime tropical maritime air. It is apparent that near the ground
bending departures reflect refractive index profile departures from stand-
ard. Figure 3.16 presents a series of graphs of departures of refractive
^
3.1
1.9
1.7
1.5
1.3
0.9
0.7
0.5
.Cold front
4 posses Rtcttfnond
I Squoll line
I crosses pothlCold front posses Wash.
I230 1650 20SO 0030
July 19, 1947
8.
10-
12 Cold front passes
y*tos^lngtool830
— July 19-
+7 Squ oil lino crosses
potti 1330 JolyO
-3
Cold fron t posses RKJimond
15" _4 2130 July 19
-2
Points 6, 1 3, a 14 missinij
320 330 340 350 360 370 380 390
Hourly Mean Surface Refracfivify
Figure 5.4. Field strength and N changes during frontal passage.
Scatter diagram showing passage of a cold front system over the Washington-Richmond path and graph
indicating the fluctuation of hourly median field strength with time.
180 SYNOPTIC RADIO METEOROLOGY
index and bending from normal for each air mass and emphasizes the
close relation between the two. This affords the synoptic radio meteor-
ologist a set of standard reference profiles for the study of a given air mass
or the confluence of contrasting air masses at a frontal zone.
Arvola [24] discussed the changes in refractive index profiles caused by
migratory weather systems. Examining a series of synoptic situations in
the midwestern portion of the United States during November 1951 that
gave rise to greater-than-normal refraction, Arvola found that ridges and
accompanying subsidence effects generally gave rise to strong A^ gradients
and enhanced signal strength over a 200-km link broadcasting at 71.75
Mc/s. Refractive gradients were stronger in the warmer air masses and
at times when moist air was present below the inversion created by the
subsidence mechanism. Strong gradients which appeared behind a squall
line later weakened with the approach of a cold front. After the passage
of this front stratification in the cold air again increased the gradient.
Subsequent investigations of polar continental air across central North
America make use of reduced-to-sea-level forms of the radio refractive
index as synpotic parameters. The reduced forms are sensitive indica-
tors of synoptic changes and afford a clearer picture of storm structure
than that obtained using analyses in terms of unreduced N or B units
(defined in chapter 1). Later portions of this chapter consider in detail
these two synoptic parameters.
Jehn [25], at the University of Texas, used a form of potential refractive
modulus, K, developed by Lukes [26] and Katz [27] to account for the
height dependence feature of the refractive index. Articles by Jehn
[28, 29] on synoptic climatology use composite analysis techniques to
study the synoptic properties of the Texas-Gulf cyclone and the central
United States type of cold outbreak.
In another application of the potential refractive modulus, Flavell and
Lane [30] have utilized Katz's K unit (see chapter 6) to study tropospheric
wave propagation. Field strength measurements on VHF-UHF trans-
horizon radio links over the British Isles are analyzed in terms of cross
sections in terms of K and regional charts of AK, the difference between
K at the surface and /^gsombar- These charts show features similar to
those obtained by use of No or A (see chapter 1).
The authors cite a series of measurements on a 500-km path at 877
Mc/s on which the received signal was ordinarily below the noise level.
The singular occasions on which the received signal could be measured at
the normal times of radiosonde ascent all exhibited a symmetric variation
of AK over the transmission path. These results lend credence to the
hypothesis that synoptic disturbances play an important role in trans-
horizon telecommunicat ions.
Moler and Arvola [31] advanced the hypothesis that the vertical
gradient of the refractive index is affected by broad-scale vertical motion
BACKGROUND
181
SAN JUAN, W.I.
(July Average)
310 320 330 340 350 360 370 380 390
A(h,3l3), N UNITS
I I I I i \ I \ I
-8 -7 -6 -5 -4 -3 -2 -l
T313-T (mrad)
Figure 5.5. Departures of t and N from normal for maritime tropical air.
182 SYNOPTIC RADIO METEOROLOGY
in the troposphere and suggest that moisture and temperature stratifica-
tion is modified principally by changes in vertical velocity. This latter
work was extended [32] by a study of mesoscale centers of horizontal air
mass convergence and divergence in the troposphere. Horizontal con-
vergence takes place in a low-pressure area, where the winds around the
low have a predominant component toward a local vortex at the center.
Upward vertical motion results from the pile-up of air in the vortex region.
In a high-pressure area, winds have a predominant component away from
the center of a high and subsiding air descending from higher levels takes
the place of air transported outward from the center of the high. Local
convergence, then, implies upward vertical motion in the lower levels of
the troposphere, while local divergence implies downward vertical motion
in the lower levels. It follows that local convergence centers (small-
scale low-pressure cells) produce updrafts in the atmosphere that result in
considerable mixing and the destruction of atmospheric layers. Centers
of divergence (small-scale high-pressure cells) create strong temperature
and humidity inversions by the motion of subsiding air. Such inversions
produce large vertical refractive index gradients that partially reflect
microwaves traveling through this meteorological environment [33]. In
a well-mixed atmosphere, on the other hand, the primary propagation
mechanism is believed to be scattering by turbulent fluctuations of the
refractive index [34, 35].
Sea-level measurements showed N s to be essentially invariant during
the experiment, yet signal levels ranged over more than 60 dB, a power
factor on the order of 10^ Since scattering theory would account for a
rise in signal level of only about 13 dB, Moler and Holden [32] conclude
that refractive layering and thermal stability over the oceans are princi-
pally functions of vertical wind velocities. These conclusions bear out
earlier ones of Saxton [36] and later ones of Flavell and Lane [30] stating
that high signal levels from a distant transmitter may well be the con-
sequence of refractive layering and subsequent reflection of radio waves.
In the same article Saxton also considered both the scattering of radio
energy by turbulent eddies and the effects of superrefraction of radio
waves produced by departures from normal of the height variation of the
tropospheric refractive index.
A knowledge of the vertical motion of the atmosphere becomes at this
point central to the problem of refractive layering. A brief resume of the
Moler-Holden method for estimating the relative magnitude and direction
of the vertical component of the wind velocity follows.
Moler and Holden postulate a model atmosphere bound by the follow-
ing conditions:
BACKGROUND 183
(1) A barotropic level (level of nondivergence) exists at some pressure
level greater than 500 mbar.
(2) The horizontal wind velocity divergence changes sign at the level
of nondivergence (LND).
(3) The vertical velocity vector in the troposphere is proportional to
that at the LND.
(4) The vertical velocity vector (a) vanishes at the surface of the earth
and (b) approaches zero at the outer reaches of the atmosphere.
Commencing with the equation of continuity,
l^+F-Vp + pV-F + l (pW) = 0, (5.1)
where p is the density of air, V is the horizontal del operator, V Is the
horizontal wind velocity vector, W is the vertical wind speed, and z is the
vertical coordinate; and employing horizontal velocity divergence in the
natural coordinate system of the form
F = f^ + . f . (5.2)
ds dn
Moler and Holden derive for the magnitude of the vertical vector at the
LND, upon employing the vorticity equation.
(TF)lnd =
p
^ (I) (11)
1 ""
_7LND ^ dt J LND S dS
(III)
LND i OZ J
(5.3)
where f is the vertical component of the absolute vorticity. Moler and
Holden reason that of the labeled integrals (I) decreases in proportion
to the height above the LND and is « (II) at 300 mbar. Since W « V,
(III) makes only a small contribution. The expression for (IF)lnd be-
comes
W.No = - i f" Vf^(f-f)dz, (5.4)
P 7 LND f ds \dx dy/
184
SYNOPTIC RADIO METEOROLOGY
Figure 5.6. Isobaric and streamline maps.
which simplifies to the approximate relation
(W) = relative magnitude of — — { T~ — ^:~ )
f ds \dx dy/s
for purposes of calculation.
(5.5)
REFRACTIVE INDEX PARAMETERS
185
— ou
-60
-70
/
\
/
A
\
^
^
—80
09
10
11
12
PST
13
14
15
Figure 5.7. X-band signal level versus time.
Santa Barbara X-band signal level for 26 Mar. 1958.
Moler and Holdeii then continue with a description of the large signal
enhancement and deep fading that occur as the propagation mechanism
varies between partial reflection and scattering on a transhorizon radio
path along the California coastline. Reflection occurs when strong re-
fracting layers are present within a kilometer of the surface, typifying
meteorological conditions generally associated with the subsidence inver-
sion frequently found along the California coast.
Figure 5.6 shows the sea level pressure chart and a series of streamline
analyses by Moler and Holden depicting mesoscale centers of convergence
and divergence for a day in March along the southern California coast.
Figure 5.7 shows the X-band signal level received at Point Loma, San
Diego (SD) from Santa Barbara (SB) during the same day. As the
centers of convergence along the radio path weaken, refractive layers are
formed and the signal level rises sharply during the middle of the day.
Later the signal level lowers again with the regeneration of convergence
centers and the destruction of stratified layers during the afternoon hours.
5.3. Refractive Index Parameters
In later sections, the analysis of a synoptic disturbance in the tropo-
sphere will be described in detail. Certain reduced forms of the refrac-
tive index that will be useful in the ensuing discussion will be developed
here. These forms, already discussed in chapter 1, are revisited here for
the purj)ose of com])arison in synoptic example.
186
SYNOPTIC RADIO METEOROLOGY
Figure 4.1 shows contours of the mean value of A'^ at the surface, N s,
determined from eight years of data for August, 0200 local time. Minia-
ture circles indicate the 62 observing stations used to analyze this chart.
It is evident that coastal areas display high values of A'^s as compared with
inland locations. Low values of N s are apparent along the Appalachian
mountain chain and in the great mountain systems and inter-mountain
plateaus of the western United States. There is a marked similarity be-
tween the N s contours on figure 4.1 and the elevation contours of figure
5.8. As a sensitive indicator of changes in atmospheric density, A''^ dis-
plays a strong elevation dependence. To remove this effect the reduced-
to-sea-level expression, A^'o, was introduced in chapter 4 as:
No = Ns exp
H*
(5.6)
where z is height in kilometers and H* = 7.0 km is the scale height.
Scale height is the height at which the mean value of A^ has decreased to a
fraction 1/e of its initial value. A scale height of 7.0 km is in close agree-
ment with H* = 7.01 km for the NACA standard atmosphere with 80
percent relative humidity and H* = 6.95 km obtained from climatic
studies utilizing over two million observations of the variation in A^ over
the first kilometer above the surface of the earth [20]. The A^'o contours
'00&500
7000 a ovERjasa
Figure 5.8. Ground elevation above sea level.
REFRACTIVE INDEX PARAMETERS
187
Figure 5.9. Idealized diagram of a fast-moving cold front.
of figure 4.4 utilize the same data as figure 4.1. The use of A^o produces a
simpler map with a smaller range of variation. Additionally, N ^ may
easily be estimated from the smooth and slowly varying contours of A^'o
providing only that station elevation is known. It was shown in chapter
4 that N s may be more accurately estimated from charts of A^o than
from charts of N ^ itself by a factor of 4 or 5 to 1 [37].
The attempt to find a workable method to compensate for the decrease
of A'' with height has brought about the development of various model
atmospheres discussed in chapter .3. The paragraphs that follow will
outline briefly steps in this development that are relevant to synoptic
studies.
Vertical refractive index cross sections are standard working charts for
synoptic studies. Such charts constructed from observed values of N
suffer from a serious shortcoming in that the natural decrease of A'' with
respect to height effectively masks contrasts between air masses in the
lower troposphere. An idealized synoptic example depictmg the con-
fluence of contrastmg air masses is presented on figures 5.9 and 5.10.
When these idealized systems are analyzed in terms of A'^ as on figures 5.11
and 5.12 the most prominent feature is the laminar structure of the A'^
field.
188
SYNOPTIC RADIO METEOROLOGY
LEGEND
WARM FRONT — -^^
ISOTHERM
RH ISOPLETH
CLOUD PATTERN
PRECIP PATTERN ^^^§
AIR FLOW
100 200 300 400 500 600 700 800 900
DISTANCE, km
Figure 5.10. Idealized diagram of a warm front.
-500
-400 -300 -200 -100 100
DISTANCE IN KILOMETERS
200
300
Figure 5.11. Idealized cold front in N units.
REFRACTIVE INDEX PARAMETERS
189
LEGEND
WARM FRONT
N ISOPLETH
300 400 500
DISTANCE, km
Figure 5.12. Idealized warm front in N units.
Early attempts to compensate for the decrease of A^ with height used
the constant gradient of the effective earth's radius theory, l/4a, where a
is the radius of the earth. As an illustration, the strong elevated layer
found during the summer in southern California was studied in terms of a
form of (1.32) for B, given by
B = N{z) + (39.2)2,
(5.7)
where N{z) is the value of A^ at height z in kilometers [38]. Since A^ tends
to be an exponential function of height rather than the linear function
assumed by the effective earth's radius theory, the B unit approach over-
corrects when z is greater than about 1 km.
This point is illustrated by figures 5.13 and 5.14 where the A'^ data of
figures 5.11 and 5.12 are plotted in terms of B units. Note that the over-
correction produces a cross section in which A'^ increases with height from
a value of 310 at the surface to 360 at 5 km. A function of exponential
form was designed to account for the systematic decay of density with
height that characterizes the terrestrial atmosphere, as given by
A = A^(z)+313[l - jexp {- f^\']
(5.8)
190
SYNOPTIC RADIO METEOROLOGY
-500 -400 -300
-200 -100 100
DISTANCE IN KILOMETERS
200 300
Figure 5.13. Idealized cold front in B units.
The quantity, A, enables one to discern departures of A'^ structure from
the model atmosphere
N = 313 exp \-f^\
Further, the radio-ray bending,
(5.9)
Tl,2 =
iV,
N.
cot e dN ' 10~',
(5.10)
where 6 is the local elevation angle of the radio ray to spherically stratified
surfaces of constant A'^, may be approximated by
Tl.2 = —
"^ 'eot d (IQ-^)
Ai n
dA{z, 313) + t(z, 313). (5.11)
The term t{z, 313) is the bending in the average atmosphere given by (5.9),
while the integral term represents the departures in bending produced by
REFRACTIVE INDEX PARAMETERS
191
300 400 500 600
DISTANCE IN KILOMETERS
Figure 5.14. Idealized warm front in B units.
various synoptic and air mass effects. The values of bending in the
average atmosphere are tabulated [39] and approximate methods of calcu-
lating the integral term in (5.11) to within a few percent have been given
[40].
The next logical step is to plot the frontal cross sections, previously
analyzed in terms of A^ and B, in A units. This has been done on figures
5.15 and 5.16. The range of refractivity values on the new charts is re-
duced from more than 60 to about 25 units and a pattern emerges that
displays sharp contrasts for air mass differences associated with the
frontal zone. Note, for the warm front case (fig. 5.16), that the A values
increase with height until they reach a maximum associated with the up-
gliding warm moist air overriding the frontal surface. The region of
precipitation in advance of the front is shown as an area of high surface
A^. In the cold-front case (fig. 5.15), the classic push of warm air aloft
by the encroaching cold air is evidenced by the "dome" of high A values
just before the front. Stratification in the cold air due to inversion
effects, although impossible to detect in the A'^ charts, is clearly seen by
the use of A unit charts.
192
SYNOPTIC RADIO METEOROLOGY
-200 -100 100
DISTANCE IN KILOMETERS
Figure 5.15. Idealized cold front in A units.
A units were used in subsequent cross-section analyses in order to
throw frontal discontinuities and air mass differences into sharp relief.
The Potential Refractivity Chart of figure 5.17 facilitates the rapid con-
version of A^ to A. This simplification eliminates the necessity of using
exponential tables for each individual calculation of A and thus lends con-
siderable ease to the preparation of charts of the new parameter.
The reader probably has already observed that the A^'o and A correc-
tions do substantially the same thing. Their primary distinction is that
^ is a nonlinear "add-on" correction while A^o is a multiplicative one.
The disparity between A^o and A is tabulated in table 5.1.
These figures are obtained by taking the zero values of A^ (for example,
300), subtracting the add-on correction for, say, 3 km (300 — 109 = 181),
and reducing this number, A'' = 181, to zero elevation by the A^o reduction.
No = N exp (3/7). The 313 exponential atmosphere is adopted for a
single reference atmosphere. The large discrepancies of table 5.1 may be
avoided for practical applications by choosing a model near to the mean
value of A^ of the site under study.
The A unit has the additional advantage that, while it is a convenient
method for height reduction, the ray bending is also readily recoverable
REFRACTIVE INDEX PARAMETERS
193
8-
LEGEND
WARM FRONT
A=N + NoCl-exp(-I)j
WHERE No=3l3
H=70 km
300 400 500
DISTANCE, km
600
Figure 5.16. Idealized warm front in A units.
from it, requiring only a knowledge of the altitude of the observed refrac-
tive index measurement and the local elevation angle of the radio ray.
The potential refractive modulus of I. Katz [27],
<P
Po + b
eo
(5.12)
where 6 is the potential temperature and eo the potential vapor pressure,
is also in current use. The constants b and c of (5.12) are given in the
development of modification to A'' data in chapter 1 (1.39). The potential
refractive modulus has been employed by Jehn [25] to study polar waves
over North America. Refractivity, A'', cannot accurately be recovered
from (p for bending calculations unless additional information is available;
namely, observed temperature and vapor pressure. The concept of the
potential refractive modulus arose out of the earlier refractive modulus,
M (see ch. 1), which may be defined by
M
4
1 +
a_
X 10' = N(z) +
z
Laj
X 10'=
(5.13)
where z = height above the earth's surface and a — earth's radius.
194
SYNOPTIC RADIO METEOROLOGY
A=N+Nol[-exp(chl]
WHERE No=3l3
c =-.143859
h=ht. in km
170 190 210 230 250 270 290 310 330 350 370 390 410 430
\6
N=(n-l)IO'
Figure 5.17. Potential refractivity chart.
Table 5.1. Differences between No and A at various elevations
Value of N at z=0
Elevation
250
300
313
350
400
(km)
1
-10
-21
-49
-48
-2
-5
-22
-29
5
7
5
30
13
2
3
4
29
31
68
A SYNOPTIC ILLUSTRATION 195
The M unit came into being out of an approach similar to that which
led to the development of the B unit. The condition dn/dz = — 1/a (a
radio duct) implies an effective earth of infinite radius (effective earth's
radius factor, /v = oo, see ch. 1). The M unit is designed so that dM /dz
= when k — oo. M units are employed from time to time in radio
meteorological analysis. The Canterbury Project [41], for example, used
M unit analyses in the study of ranges of over-water radar signals.
5.4. A Synoptic Illustration
The specialized field of synoptic radio meteorology attempts a descrip-
tion of the variations in atmospheric refraction that arise from large
scale weather changes such as the passage of a polar front or the move-
ment of an air mass over a particular geographic region. The term air
mass is used to describe a portion of the troposphere that has at the surface
generally homogeneous properties. Although no air mass is in fact
homogeneous, the advantages of the air mass concept as a convenient
fiction are evident in the cataloging of meteorological observations for
climatic or synoptic purposes.
The region of interaction between the cold air of the poles and the
warm air of the tropics is referred to as the polar front and is generally
located between 30 and 60°N. From time to time a section of the polar
front is displaced northward by a flow of warm tropical air while an adja-
cent section is simultaneously displaced southward by a flow of polar
air. The interaction of the flow of polar and tropical air results in the
formation of a "wave" that moves along the polar front, often for thou-
sands of kilometers. An example of a fully developed polar front wave is
shown on figure 5.18(a), in the same manner that it would appear on a
daily weather map. Across the Great Plains and eastern seaboard of the
United States the polar front wave normally moves along the line AB in
figure 5.18(a). An idealized space cross section along the line AB is
shown in figure 5.18(b). The warm tropical air that flows into the warm
sector of the wave overrides the cool air before the wave to form the
transition zone denoted as a warm front. The cold front represents the
transition between the generally humid air of the warm sector that has
been forced upwards and the advancing cold polar air. Squall lines are
drawn to represent belts of vigorous vertical convection, intense thunder
showers, and sharp wind shifts that frequently precede fast-moving cold
fronts. The fronts shown on a daily weather map represent the ground
intersection of the transition zones between various air masses.^
2 The reader who wishes a critical appraisal of recent meteorological thinking on
fronts, air masses, squall lines, etc., is referred to Dynamic Meteorology and Weather
Forecasting, by Godske, Bergeron, Bjerknes, and Bundgaard, American Meteoro-
logical Society and Carnegie Institute of Washington, D.C., 1957.
196
SYNOPTIC RADIO METEOROLOGY
Figure 5.18. The Polar Front wave.
SURFACE ANALYSIS IN TERMS OF No 197
As a representative example of the application of these newly developed
units to a synoptic situation, a large-scale outbreak of continental polar
air which took place over the United States during February 1952 is
analyzed in terms of A^o and A [42, 43].
For this synoptic illustration the reduced expression, No, is used in
preparing constant level charts of the storm at the same levels and times
as those used in the daily weather map series of the U.S. Weather Bureau.
The A unit, on the other hand, is employed to construct vertical cross-
sections through the frontal system to give a three-dimensional picture of
the synoptic changes taking place.
5.5. Surface Analysis in Terms of N^
A pronounced cold front developed and moved rapidly across the
United States during the period 18 to 21 February 1952.
Prior to February 18, a polar maritime air mass had been moving slowly
eastward across the Great Basin and Rocky Mountain regions. This
system included a slow-moving cold front extending from northern Utah
southwards into Arizona and a quasi-stationary front extending north-
eastwards into Wyoming. With the outbreak of polar continental air
east of the Rocky Mountains, the maritime front became more active
and, as it moved ahead of the fast-moving polar continental front sweep-
ing across the Great Plains, was reported as a squall line by the time it
crossed the Mississippi River early on the morning of February 20.
During the latter stages of the storm system, the polar maritime cold
front-squall line was located in the developing warm sector of the polar
front wave. The entire ensemble of cold front, polar front wave, and
squall line then moved rapidly to the east coast by the morning of the
21st, thus completing the sequence.
Charts of Ao were prepared from Weather Bureau surface observations
taken at 12-hour intervals from 0130 EST, February 18 until 0130 EST,
February 21, 1952, or, in other words, the period of time that it took the
polar front wave to develop and move across the country. The synoptic
sequence is seen on figures 5.19 through 5.25., where contours of A^o are
derived for various stages of the storm and compared to the superimposed
Weather Bureau frontal analysis. The same procedure of comparing
derived contours with the existing frontal pattern was followed throughout
the present example. Observations from 62 weather stations were used
in preparing the surface weather maps. Figure 5.19 indicates that the
cold front extending from Utah southward displays weak A^o changes
across the frontal interface. In the early stages of the sequence (figs.
5.19 and 5.20) this lack of air mass contrast is evidenced in another way
by the slight change of the position of the Ao = 290 contour encircling
west Texas and New Mexico as the frontal system moves through that
area.
198
SYNOPTIC RADIO METEOROLOGY
Figure 5.19. No chart for storm system OlSOE 18 Feb. 1952.
Figure 5.20. No chxirl for storm system 1S30E 18 Feb. 1952.
SURFACE ANALYSIS IN TERMS OF A^o
199
Figure 5.21. No chart for storm system 0130E 19 Feb. 1952.
Figure 5.22. No chart for storm system 1S30E 19 Feb. 1952.
200
SYNOPTIC RADIO METEOROLOGY
Figure 5.24. No chart for storm system 1330E 20 Feb. 1962.
SURFACE ANALYSIS IN TERMS OF No
201
Figure 5.25. No chart for storm systems 0130E 21 Feb. 1952.
By comparison, the cold front sweeping down across the Great Plains
(figs. 5.20 to 5.22) has a rather marked A^o gradient across the front, due
in large measure to the northward flow of warm, moist air that forms a
definite warm sector by 1330 EST on the 19th. It is perhaps significant
that the A^o contours indicate that the various frontal systems are transi-
tion zones rather than the sharply defined discontinuities of textbook
examples, a point that has been enlarged upon by Palmer [44].
Figures 5.22 to 5.25 trace the trajectory of this vigorous push of cold
air across the Gulf Coastal Plain and the southeastern states. The most
spectacular gradients on this map series are in the eastern half of Texas
where the marked contrast of cold, dry polar air of low A'^o and warm
moist Gulf air of very high A^o occurs. A prominent feature of meteor-
ological significance on figures 5.20 to 5.23 is the northward advection of
tropical maritime air in the warm sector ahead of the cold front. The
advection is evidenced by the northward bulge of high A^o over the
Mississippi Valley on both charts. Figure 5.25 shows the synoptic situa-
tion as the front moves off into the Atlantic Ocean and refractivity
gradients across the continent gradually weaken.
The variation of A'^o due to the passage of the frontal system can be
seen on figures 5.26 to 5.30 where the 24-hour changes of A^o have been
contoured. The 24-hour change, designated AVo, is obtained by sub-
tracting from the current value the value of A''o observed 24 hours ago.
202
SYNOPTIC RADIO METEOROLOGY
Figure 5.26. 24-hour ANo chart, OlSOE 19 Feb. 1952.
Figure 5.27. 24-hour ANo chart, 1330E 19 Feb. 1952.
SURFACE ANALYSIS IN TERMS OF No
203
Figure 5.28. S4-hour ANo chart, 0130E SO Feb. 1962.
Figure 5.29. 24-hour ANo chart, 1330E 20 Feb. 1962.
204
SYNOPTIC RADIO METEOROLOGY
Figure 5.30. U-hour ANo chart, 0130E 21 Feb. 1952.
The change is determined on a 24-hour basis in order to remove effects of
the diurnal cycle of A^'o. The AA^o charts show a general rise of Nq in the
warm sector and a drop in A^^o behind the front amounting, in the warm
sector, to 35 to 40 A^ units by 1330 EST on February 20 (fig. 5.29) accom-
panied by a 40 to 50 A^ drop behind the front.
The relative sensitivity of A^o to humidity changes is emphasized by
the AA^'o charts. The N drop behind the cold front occurs in a region of
increasing pressure and decreasing temperature — a combination that
increases the dry term and depresses the wet term. The decrease in the
wet term from rapidly dropping dewpoint more than compensates for the
increased dry term. As an example, in the 24-hour period ending 0130
EST on the 19th, the station pressure at Oklahoma City increased 13 mbars.
The dry term increased 12 N units while the wet term dropped 42 A'^ units,
giving a net change of minus 30 A'' units. This A^'o rise in the warm sector
and the drop behind the cold front is consistent throughout the develop-
ment of the polar front wave and appears to be what one would expect
for this type of weather system. The present system had about a 35 A^
unit rise and a 40 A?^ unit drop. This general pattern might be expected
to occur in all fast-moving cold fronts with varying intensity, depending
upon the individual synoptic pattern. In any case, it appears that the
Nq pattern is a sufficiently stable and conservative property of the atmos-
phere so that it should be possible to develop forecasting rules for A^^o,
but not, of course, without the analysis of many more N patterns.
CONSTANT PRESSURE CHART ANALYSIS 205
5.6. Constant Pressure Chart Analysis
The same frontal system as above was analyzed for selected constant
pressure levels. The 850 mbar charts (about 5,000 ft above mean sea
level) and the 700 mbar charts (about 10,000 ft above mean sea level)
were prepared for the times of radiosonde ascent (10 A.M. and 10 P.M.
EST) throughout the synoptic sequence from the radiosonde reports of
43 U.S. sounding stations. It is not necessary to reduce the 850 mbar
or 700 mbar level data, since it is already referenced to the indicated
constant pressure level. Contours for the charts aloft are shown on
figures 5.31 to 5.36 while their respective 24-hour changes, Nsbo and A^too,
are given on figures 5.37 to 5.40.
The Nsbo charts show that the northerly flow of warm humid air within
the w^arm sector that was so prominent on the A^o maps is also clearly in
evidence at the 850-mbar level. Further, a change pattern similar to
that on the A'o maps is also observed at the 850-mbar level, particularly
on figure 5.34. That is, a rise in Agso values in the w^arm sector and a
decrease behind the cold front is apparent. Surprisingly enough, by the
time the frontal system is well-developed, at 1000 EST on the 20th, the
Agso values are nearly as large as those on the surface.
The A'voo charts are more difficult to interpret than those of A^o or
A'sso- It appears that at this altitude the wet term is usually negligible
and N will normally vary inversely as temperature since the pressure is,
of course, constant at the 700-mbar level. By 1000 EST on the 19th (fig.
5.34) an intrusion of low A" values is observed in the 700-mbar warm sector
due to the advection of warm, low-density air northwards. The chart
for 24 hours later (fig. 5.42) displays two prominent highs in which
A700 = 225. One of these highs lies between the squall line and the cold
front and the other just south of the apex of the 700-mbar wave. Inter-
estingly enough, these two highs are due to quite different causes. The
high centered over Atlanta appears to have arisen from the unusually high
transport of moisture to the 10,000-ft level, since the 700-mbar wet term
at Atlanta increases from 4.5 to 25 N units in the 24-hour period ending
with 1000 EST on February 20. The second high, centered over Omaha,
appears to be due to an intense dome of cold air, as indicated by the drop
of the 700-mbar temperature from -7.3 °C to -21.4 °C in 12 hours
preceding map time. When temperatures are below °C, the wet term
contribution to A^ is quite small and density changes become significant
in producing changes in A. Falling temperatures produce higher density
air and, consequently, a region of high A^ values around Omaha as de-
picted on the A^voo chart of figure 5.31 and AA700 chart of figure 5.40 which
shows this change more clearly.
206
SYNOPTIC RADIO METEOROLOGY
Figure 5.31. Nsso chart, lOOOE 18 Feb. 1952.
Figure 5.32. N700 chart, lOOOE 18 Feb. 1962.
CONSTANT PRESSURE CHART ANALYSIS
207
Figure 5.33. Ngso chart, lOOOE 19 Feb. 1952.
Figure 5.34. N700 cMrt, lOOOE 19 Feb. 1962.
208
SYNOPTIC RADIO METEOROLOGY
Figure 5.35. Nsso chart, lOOOE 20 Feb. 1952.
Figure 5.36. N700 chart, lOOOE 20 Feb. 1952.
CONSTANT PRESSURE CHART ANALYSIS
209
Figure 5.37. 24-hour ANgso chart, lOOOE 19 Feb. 1952.
Figure 5.38. 24-hour AN700 chart, lOOOE 19 Feb. 1962.
210
SYNOPTIC RADIO METEOROLOGY
Figure 5.39. 24-hour ANsbo chart, lOOOE 20 Feb. 1952.
Figure 5.40. 24-hour AN 700 chart, lOOOE 20 Feb. 1962.
VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX
211
5.7. Vertical Distribution of the Refractive
Index Using A Units
The synoptic study of the vertical distribution of the radio refractive
index extends the foregoing constant-level analyses by considering the
problem of whether the air mass properties associated with this typical
wintertime outbreak of polar air are reflected in the vertical refractive
index structure. Charts showing the structure of the storm have been
prepared using radiosonde measurements from stations located along a
line normal to the frontal zone between Glasgow, Mont., and Lake
Charles, La. (fig. 5.41). Plots of A'^ versus height along this cross section
line were obtained at 12-hour intervals during a 4-day period and con-
verted to A units.
Figure 5.42 is an example cross section along the Glasgow-Lake Charles
line analyzed in terms of unmodified A'' as in the idealized cases of figures
5.11 and 5.12. Compare this figure with the A unit analysis of figure
5.48 for the highlighting of air mass differences refractive-index-wise.
Examples of the distribution of A'^ components, temperature and humidity,
around the front are charted on figures 5.43 and 5.44. Various stages
in the advance of this intense storm system across the continent are
represented by figures 5.45 to 5.50 in terms of modified A'^ (A units). At
the outset of the period of observation (fig. 5.45), the polar front was
located over the northern Great Plains, between Rapid City, S. Dak.
Figure 5.41. Station identificalion chart.
212
SYNOPTIC RADIO METEOROLOGY
1000 1500
DISTANCE IN KILOMETERS
GGW RAP LBF DDC OKC LIT
Figure 5.42. Space cross section in N units, 1600Z, 19 Feb. 1962.
2300
LCH
DISTANCE IN KILOMETERS
GGW RAP LBF DDC OKC LIT
Figure 5.43. Temperature cross section in °C, 1500Z, 19 Feb. 1952.
LCH
VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX 213
and North Platte, Nebr. At this time the entire cross section is charac-
terized by weak to moderate gradients of refractivity. On figure 5.46
12 hours later, this front had moved some 300 km southward. The
contrast of the southward push of polar air and the northerly advection
of tropical maritime air from the Gulf of Mexico into the developing warm
sector of the polar front wave is evidenced by the relatively large gradients
in the neighborhood of Dodge City. The core of tropical maritime air has
evidently not progressed far enough northward to displace the warm but
dry air that had been over the Great Plains prior to the outbreak, with
the result that a region of low A values is found between the front and the
tropical maritime air. On figure 5.47 the effects of the Pacific front are
apparent in the buildup of a secondary region of high A some 400 km
ahead of the polar front, located at this time over Dodge City. Twelve
hours later (fig. 5.48) the core of tropical maritime air has become more
extensive and now reaches to a height of 3 km. The second (Pacific)
front is picked up now on the cross section and the area of low A values is
confined between the two fronts. By 0300 UT on the 20th, figure 5.49,
the polar front is approaching Lake Charles and the Pacific front is re-
ported on the daily weather map as a squall line. Finally, by the morning
of February 20 (fig. 5.50), both fronts have passed to the south of Lake
Charles, and the polar air just behind the front is characterized by rela-
tively low A values.
The use of space cross sections does not always yield measurements
at the most desirable points along a frontal zone. Another method of
arriving at the probable refractive index structure about the frontal
interface is to plot radiosonde observations for a single station arranged
according to observation times as on figures 5.51 to 5.57.
The time cross section for Glasgow, Mont. (fig. 5.51), which is in the
cold air behind the polar front for the entire period of observation of the
storm, displays gradients of A values that are generally weak. An excep-
tion occurs on the evening of the 19th (20/0300 UT) apparently as a
result of subsidence effects. Rapid City, S. Dak. (fig. 5.52), similar to
Glasgow, shows generally weak gradients throughout in the cold air be-
hind the front. North Platte, Nebr. (fig. 5.53), exhibits moderate
gradients with increasing A values in the post-frontal, higher-density polar
air. This station is too far north to record much in the way of moisture
effects at this time of year. Dodge City, Kans. (fig. 5.54), represents
dry low ahead of the polar front and increasing A values in the cold air
just behind the front. Oklahoma City, Okla. (fig. 5.55), represents a
classic synoptic situation in which advection of tropical maritmie air
from the Gulf of Mexico produces a strong high ahead of the front and a
low within the cool, dry, polar continental air. Again in the region
around Little Rock, Ark. (fig. 5.5G), there is warm, moist air of extremely
high refractivity ahead of the front being replaced by polar continental
214
SYNOPTIC RADIO METEOROLOGY
1000 I 1500
DISTANCE IN KILOMETERS
2300
GGW RAP LBF DDC OKC LIT LCH
Figure 5.44. Relative humidity cross section (percent), 1500Z, 19 Feb. 1952.
DISTANCE IN KILOMETERS
I I
GGW RAP LBF DDC OKC LIT
Figure 5.45. Space cross section in A units, 0300Z, 18 Feb. 1962.
LCH
VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX 215
2000 2500
DISTANCE IN KILOMETERS
DODGE CITY
GLASGOW DODGE CITY LAKE CHARLES
Figure 5.46. Space cross section in A units, 1500Z, 18 Feb. 1962.
3.0
1000 I 1500
DISTANCE IN KILOMETERS
GGW RAP LBF DDC OKC LIT LCH
Figure 5.47. S-pace cross section in A units, 0300Z, 19 Feb. 1962.
216
SYNOPTIC RADIO METEOROLOGY
2000 2300
DISTANCE IN KILOMETERS
GLASGOW DODGE CITY LAKE CHARLES
Figure 5.48. Space cross section in A units, 1500Z, 19 Feb. 1962.
1000 I 1500
DISTANCE IN KILOMETERS
GGW RAP LBF DDC OKC LIT
Figure 5.49. Space cross section in A units, 0300Z, 20 Feb. 1962.
2300
LCH
VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX 217
DISTANCE IN KILOMETERS
I
GLASGOW DODGE CITY LAKE CHARLES
Figure 5.50. Space cross section in A units, 1500Z, 20 Feb. 1962.
I80300Z I8I500Z I90300Z
I9I500Z 200300Z
DAY AND HOUR
20I500Z 2I0300Z 2II500Z
Figure 5.51. Time cross section, Glasgow, Mont., in A units.
218
SYNOPTIC RADIO METEOROLOGY
3.0
2.5
2.0 —
1.5 —
0.5
I80500Z I8I500Z
/ \
1 /
' X
y \
/ /
^^«^
i;^^ Vj /
r"\
U ^
^
r\
\
^ ^315—^
//3?0^
— —
I90500Z
I9I500Z 200300Z
DAY AND HOUR
20I500Z 2I0300Z
2II500Z
Figure 5.52. Time cross section, Rapid City, S. Dak., in A units.
3,000
I80300Z
20I500Z
I8I500Z I90300Z I9I500Z 200300Z
DAY AND HOUR
Figure 5.53. Time cross section, North Platte, Nebr., in A units
2I0300Z 2II500Z
VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX 219
180300Z iSISOOZ I90500Z I9I500Z 200300Z 20I500Z 2I0300Z 2II500Z
DAY AND HOUR
Figure 5.54. Time cross section, Dodge City, Kans., in A units.
3,000
2,500
O
I80300Z I8I500Z I90300Z I9I500Z 200300Z 20I500Z 2I0300Z 2II500Z
DAY AND HOUR
Figure 5.55. Time cross section, Oklahoma City, Okla., in A units.
220
SYNOPTIC RADIO METEOROLOGY
I80300Z I8I500Z I90300Z I9I500Z 200300Z 20I500Z 2I0300Z 2II500Z
DAY AND HOUR
Figure 5.56. Time cross section, Little Rock, Ark., in A units.
3,000
I80300Z
I8I500Z
I90300Z
I9I500Z 200300Z
DAY AND HOUR
20I500Z
2I0300Z 2II500Z
Figure 5.57. Time cross section, Lake Charles, La., in A units.
VERTICAL DISTRIBUTION OF THE REFRACTIVE INDEX
221
-2700 -2500
DISTANCE FROM FRONT IN KILOMETERS
Figure 5.58. Epoch chart, Glasgow, Mont., in A units.
air of characteristically low A value. The time cross section for Lake
Charles, La. (fig. 5.57), is complex but represents again the same general
features: high A values ahead of the front and low ones behind.
The exponential correction to the refractive index height distribution
used in this storm series allows air mass properties to be clearly seen. By
use of such an exponential correction, one may construct an idealized
refractive index field about a frontal transition zone that shows the tem-
perature and humidity contrasts of the different air masses. Further,
when this technique is applied to the analysis of a synoptic tropospheric
disturbance, it does indeed highlight air mass differences.
The time cross section presentation is referred to as an epoch chart
when the observations are presented as plus or minus time deviations with
respect to the frontal passage. Thus, as a frontal system advances and
passes over a station, one obtains yet another perspective of the space
cross section. Such a presentation is given on figures 5.58 to 5.64.
Figure 5.59 represents a typical continental station located in the polar
222
SYNOPTIC RADIO METEOROLOGY
continental air mass throughout the occurrence of the storm. The essen-
tial feature here, as in figure 5.52, is the absence of detail of A structure
due to the presence of a uniform air mass over this station. Compare
this figure with the epoch chart for Oklahoma City (fig. 5.62), where the
structure of the idealized model is clearly reflected by the prefrontal A
unit high, strong gradient across the frontal zone, and the A unit low
behind the front. This rather fortuitous agreement is felt to be due to
the strategic location of Oklahoma City with respect to the motion of
contrasting air masses about the polar front. That is, this epoch chart
represents a point of confluence of virtually unmodified polar continental
and tropical maritime air.
//f //f //f //f I If //f I If ijf ijf ijf ijf ijf ijf ijf ijfijf ///- //
a5
LEGEND
A ISOPLETH
COLD FRONT
J \ L
J \ L
-2000
-1500
-500
-200
DISTANCE FROM FRONT , km
Figure 5.59. Epoch chart, Rapid City, S. Dak., in A units.
SUMMARY
223
■1600-1500
-500
DISTANCE FROM FRONT IN KILOMETERS
Figure 5.60. Epoch chart, North Platte, Nebr., in A units.
300
5.8. Summary
A survey such as this is designed to indicate the direction of radio
meteorological research. Among the brighter prospects is the work of
Moler et al., concerning the interrelation of refractive index structure and
mesoscale weather changes. Other synoptic features likely to have refrac-
tive index significance are the migratory high and low pressure cells of the
middle latitudes, since certain surface patterns and vertical profiles occur
frequently with a particular type of synoptic system.
A knowledge of the meso-macroscale behavior of A'^ in a synoptic sense
enables the propagation engineer to anticipate the occurrence of super-
refraction or ducting conditions. Thus, meteorological conditions that
give rise to such phenomena as prolonged space-wave fadeouts and inter-
ference effects from elevated layers in the troposphere can to some extent
be planned for in advance.
224
SYNOPTIC RADIO METEOROLOGY
DISTANCE FROM FRONT IN KILOMETERS
Figure 5.61. Epoch chart, Dodge City, Kans., in A units.
5.9. References
1] Sheppard, P. A. (1947), The structure and refractive index of the atmosphere,
Book, Meteorological Factors in Radio Wave Propagation, (Phys. Soc. and
Roy. Meteorol. Soc, London, England).
2] Gerson, N. C. (1948), Variations in the index of refraction of the atmosphere,
Geofis. Pura Appl. 13, 88.
3] Perlat, A. (1948), Meteorology and radioelectricity, L'Onde Elec. 28, 44.
4] Randall, D. L. (1954), A study of the meteorological effects on radio propagation
at 96.3 Mc between Richmond, Va., and Washington, D.C., Bull. Am. Meteorol.
Soc. 35, 56-59.
5] Misme, P. (1957), Influence des discontinuites frontales sur le propagation des
ondes decimetriques et centimetriques, Ann. Telecommun. 12, 189-194.
6] Hay, D. R. (1958), Air-mass refractivity in central Canada, Can. J. Phys. 36,
1678-1683.
7] Pickard, G. W., and H. T. Stetson (1950), Comparison of tropospheric reception,
J. Atmos. Terrest. Phys. 1, 32.
8] Pickard, G. W., and H. T. Stetson (1950), Comparison of tropospheric reception
at 44.1 Mc with 92.1 Mc over the 167-mile path of Alpine, N.J., to Needham,
Mass., Proc. IRE 38, 1450.
9] Bean, B. R. (1956), Some meteorological effects on scattered radio waves, IRE
Trans. Commun. Syst. CS-4, No. 1, 32.
REFERENCES
225
///-" IJF IJF IJF //F//F IJF IJF I If ijh l/h ///- /,
LEGEND
A ISOPLETHS
COLD FRONT
//
-500 500
DISTANCE FROM FRONT , km
Figure 5.62. Epoch chart, Oklahoma City, Okla., in A units.
[10] Onoe, M., M. Hirai, and S. Niwa (1958), Results of experiment of long-distance
overland propagation of ultra-short waves, J. Radio Res. Labs. 5, 79.
[11] Hull, R. A. (1935), Air-mass conditions and the bending of ultra-high-frequency
waves, QST 19, 13-18.
[12] Hull, R. A. (1937), Air-wave bending of ultra-high-frequency waves, QST 21,
16-18.
[13] Englund, C. R., A. B. Crawford, and W. W. Mumford (1938), Ultra-short-wave
radio transmission through the non-homogeneous troposphere, Bull. Am.
Meteorol. Soc. 19, 356-360.
[14] Yerg, D. G. (1950), The importance of water vapor in microwave propagation at
temperatures below freezing, Bull. Am. Meteorol. Soc. 31, 175-177.
[15] Schelleng, J. C, C. R. Burrows, and E. B. Ferrell (1933), Ultra-short-wave
propagation, Proc. IRE 21, 427-463.
[16] Misme, P. (1960), L'influence du gradient equivalent et de la stabiHt6 atmos-
ph6 rique dans les Haisons transhorizon au Sahara et au Congo, Ann. Telecom-
mun. 16, 110.
[17] Misme, P., B. R. Bean, and G. D. Thayer,(1960), Comments on "Models of the
atmospheric radio refractive index," Proc. IRE 48, 1498-1501.
[18] Gray, R. E. (1957), The refractive index of the atmosphere as a factor in tropo-
spheric propagation far beyond the horizon, IRE Nat. Convention Record,
Pt. 1, 3.
226
SYNOPTIC RADIO METEOROLOGY
-500 500
DISTANCE FROM FRONT IN KILOMETERS
Figure 5.63. Epoch chart, Little Rock, Ark., in A units.
1000 1700
[19] Gray, R. E. (1961), Tropospheric scatter propagation and meteorological condi-
tions in the Caribbean, IRE Trans. Ant. Prop. AP-9, No. 5, 492-496.
[20] Bean, B. R., and G. D. Thayer (1959), On models of the atmospheric radio
refractive index, Proc. IRE 47, No. 5, 740-755.
[21] Schulkin, M. (1952), Average radio-ray refraction in the lower atmosphere, Proc.
IRE 40, No. 5, 554-561.
[22] Fannin, B. M., and K. H. Jehn (1957), A study of radio elevation angle error due
to atmospheric refraction, IRE Trans. Ant. Prop. AP-2, No. 1, 71-77.
[23] Bean, B. R., J. D. Horn, and L. P. Riggs (1960), Refraction of radio waves at
low angles within various air masses, J. Geophys. Res. 65, 1183.
[24] Arvola, W. A. (1957), Refractive index profiles and associated synoptic patterns,
Bull. Am. Meteorol. Soc. 38, No. 4, 212-220.
[25] Jehn, K. H. (1960), The use of potential refractive index in synoptic-scale radio
meteorology, J. Meteorol. 17, 264.
[26] Lukes, G. D. (1944), Radio meteorological forecasting by means of the thermo-
dynamics of the modified refractive index, Third Conf. Prop., NDRC, pp.
107-113 (Committee on Propagation, Washington, D.C.).
[27] Katz, I. (1951), Gradient of refractive modulus in homogeneous air, potential
modulus, Book, Propagation of Short Radio Waves, pp. 198-199 (McGraw-
Hill Book Co., Inc., New York, N.Y.).
[28] Jehn, K. H. (1960), Microwave refractive index distributions associated with
the Texas-Gulf cyclone, Bull. Am. Meteorol. Soc. 41, 304-312.
REFERENCES
227
-300
500
DISTANCE FROM FRONT IN KILOMETERS
Figure 5.64. Epoch chart, Lake Charles, La., in A units.
[29] Jehn, K. H. (1961), Microwave refractive-index distributions associated with the
central United States cold outbreak, Bull. Am. Meteorol. Soc. 42, 77-84.
[30] Flavell, R. G., and J. A. Lane (1962), The appUcation of potential refractive
index in tropospheric wave propagation, J. Atmospheric Terrest. Phys. 24,
47-56.
[31] Moler, W. F., and W. A. Arvola (1956), Vertical motion in the atmosphere and its
effects on VHF radio signal strength, Trans. Am. Geophys. Union 37.
[32] Moler, W. F., and D. B. Holden (1960), Tropospheric scatter propagation and
atmospheric circulations, J. Res. NBS 64D (Radio Prop.), No. 1, 81-93.
[33] Gossard, E. E., and L. J. Anderson (1956), The effect of super-refractive layers on
50-5,000 Mc nonoptical fields, IRE Trans. Ant. Prop. AP-4, 175-178.
[34] Megaw, E. C. S. (1950), Scattering of electromagnetic waves by atmospheric
turbulence. Nature 166, 1100-1104.
[35] Booker, H. G., and W. E. Gordon (1950), A theory of radio scattering in the
troposphere, Proc. IRE 38, 401-412.
[36] Saxton, J. A. (1951), Propagation of metre radio waves beyond the normal
horizon, Proc. lEE 98, 360-369.
[37] Bean, B. R., J. D. Horn, and A. M. Ozanich, Jr. (1960), Climatic charts and data
of the radio refractive index for the United States and the world, NBS Mono. 22.
[38] Smyth, J. B., and L. G. Trolese (1947), Propagation of radio waves in the tropo-
sphere, Proc. IRE 35, 1198.
228 SYNOPTIC RADIO METEOROLOGY
[39] Bean, B. R., and G. D. Thayer (1959). CRPL exponential reference atmosphere,
NBS Mono. 4.
[40] Bean, B. R., and E. J. Button (1960), On the calculation of the departures of
radio wave bending from normal, J. Res. NBS 64D (Radio Prop.), No. 3,
259-263.
[41] Canterbury Project (1951), Vols. I-III (Department of Scientific and Industrial
Research, Wellington, New Zealand).
[42] Bean, B. R., and L. P. Riggs (1959), Synoptic variations of the radio refractive
index, J. Res. NBS 63D (Radio Prop.), No. 1, 91-97.
[43] Bean, B. R., L. P. Riggs, and J. D. Horn (1959), Synoptic study of the vertical
distribution of the radio refractive index, J. Res. NBS 63D (Radio Prop.),
No. 2, 249-258.
[44] Palmer, C. E. (1957), Some kinem.".!;ic aspects of frontal zones, J. Meteorol. 14,
No. 5, 403-409.
Chapter 6. Transhorizon Radio-
Meteorological Parameters
6.1. Existing Radio-Meteorological Parameters
6.1.1. Introduction
A method of predicting the statistical distribution of field strength on
transhorizon paths is an important requirement m tropospheric wave
propagation. Consequently, considerable attention has been given in
recent years (see figs. 6.1 to 6.6) to studies of the correlation between the
measured signal level (e.g., the monthly median value) and some quantity
derived from surface or upper air meteorological data. Figures 6.1 to 6.6
show relationships between field strength and various meteorological
parameters. It can be seen that there are some quite marked similarities
(high correlation coefficients) between the two. It thus appears that if a
reliable "radio-meteorological parameter" could be developed, then
generally available meteorological data would replace expensive radio
measurements in deriving the required distribution.
Progress has already been made in this difficult problem, [1, 2]^ In
these investigations, special attention has been given to two parameters:
(a) the surface value of refractivity, N s, and (b) the difference, AN,
between A''^ and N at a height of 1 km, Ni. Other groups have studied
different parameters [3, 4, 5, 6, 7, 8], either as possible alternatives to
N s and AN in the prediction process or as quantities which clarify the
effect of meteorological features, such as anticyclonic subsidence, on
signal strength. It is evident from the literature that some difference of
emphasis exists regarding the relative merits of the parameters proposed
to date, and particularly on the value of studies of N s-
This chapter provides a critical survey of the present position in this
field of radio meteorology, and indicates a new approach which incor-
porates some aspects of all existing treatments. Section 6.1 contains a
study of previous work and attempts to put the various views in proper
perspective; section 6.2 discusses some selected radio data from VHF
paths and its classification in terms of refractive index profiles, while
section 6.3 introduces a parameter combining the concepts of refraction
and atmospheric stability, and compares its properties with those of
existing parameters.
1 Figures in V)rackets indicate the literature references on p. 2()t).
22!)
230
TRANSHORIZON PARAMETERS
A S 6 6 H J F M A M J J A S S H
6 3 I 29 26 24 21 18 17 14 12 9 7 4 I 29 27 24 22
1947 1946
0.0
Figure 6.1. Weekly means of measured field intensities of W2WMN and W2XEA at
Needham, Mass., compared with corresponding atmospheric surface refraction at
Boston, Mass.
(After Pickard and Stetson, 1950).
1
m
A
/
^
70-
/
<.
^
■/
y
.,'
.^N
7 m dBm
.
l^
■>
-^
—
SO-
/.
/
^-
,
/<■
^
/
/ i
"'1
\
Ki—
—
^.
7
% '
y
so-
\
,,
<^
■'
„>
r^"
\
/?
/
Vi' 53,6
T..SB,2
|^= »,J2
",= -ifie
r - 0,76
b,-0.90';fi
./JT - .
\
^
^,
/)
V
•
10-
y
V
/
20-
/
R
= U,»i
>
IX —
oofnAMOLASO
Figure 6.2. Correlation of the received monthly mean field strength at Campu Sa Spina
and the gradient AN {between and 1000 m.) of the monthly median index of refraction
at Elmas.
(The values of field strength are those obtained after omission of superrefraction) . (After Bonavoglia, 1958) .
INTRODUCTION
231
1
c
■^o
^r\
n
n
1
1
\
Li^\j\j U.O.I.
i
I
■
1
\
!
\
y
\
1
1
/
\
^
\\
1
/
1
i
1
>
s
I
y^
\
-AN-
~^
1
/
\
(
/I
\
s
-^
—
"
'
\
\
.
/
{
/
\
/I
/
/
Y
/^
/
i
r
\
/
1
V
/
I
'/
\
r
—
^-
^
'
/
\
/
\
\
f
\
1
•
\
f\
1
y
\
N
\
\
Lh-
1
y
\
\
'
1
]
/
(a)
168
176
58
56
54
52
50
48
46
44
42
40
38
52
50
48
46
44
42
40
38
36
ONDJFMAMJJ ASONOJFMAMJJASOND
1950 1951 1952
Figure 6.3. Comparison of the monthly median basic transmission loss and refradivity
gradient for KIXL-FM, Dallas, Tex., recorded at Austin, Tex.
I 1
9 inn
I I
f^ CT
^
_ rr^nn n.KAT
/
1
'
/
\
'/
SI
^,
I
/
\
-AN—
.^^
/
I
i
j
I
V
J
\
*i
\
A
IJ
I
1
\
t<A
J-
,\
1
\\
Ij
\
,/
\
'
/
\ '
I
\
ll
'
\
/
\
f.
\,
y
\
1
\
jj
\
f
\
Jl
\
\
1
V
/
{
7
/
y\
1
1
\
V
/
>
1
/j
\
1
/
/
1
N
^ A
\
\
1
\\
1
i-b-
\
1
1
\
1
i^
1
1
A
—
'
(b)
178
180
182
184
X)
Q
186
c
172
_l
174
176
178
180
182
184
186
188
The development of prediction techniques is especially difficult in the
case of:
(a) any path with terminals just v/ithin or just beyond the normal
horizon, and
(b) signal enhancements which occur for small percentages of the time.
Section 6.2 therefore includes some discussion of propagation charac-
teristics on VHF paths for which considerable radio and meteorological
data are already available. The results obtained are particularly relevant
to an understanding of the large differences observed between median
232
TRANSHORIZON PARAMETERS
1945
1946
Figure 6.4. Field reception of W2XMN on 42.8 Mc/s at Needham, Mass., 1945-1946.
(After Pickard and Stetson, 1947).
signal levels on VHF paths of comparable length, frequency and angular
distance.
The primary purpose of a radio-meteorological parameter is to provide
the best estimate of the statistical distribution of field strength (in terms of
hourly, daily, weekly, or monthly median values as required on a specific
path). The reliability of the parameter must be judged solely in terms of
this requirement, and care must be exercised in assessing the value of any
given parameter in t^rms of data obtained over limited intervals of time
or from restricted geographical areas. Discussions later in the chapter
will consider to what extent it is possible to develop a parameter which,
in addition to being statistically reliable, is also characteristic of the
physical structure of the atmosphere.
The present practice in applying radio-meteorological parameters con-
sists in determining an average signal level for a given distance and then
adjusting this average level for climatic and seasonal differences by refer-
ence to the changes in some function of the refractive index of the atmos-
phere. We may express this procedure mathematically, for a linear
regression model, thus:
E = b ■ f{n) + a
(6.1)
where E is the field strength, and b is the regression coefficient expressing
INTRODUCTION
233
-1.0
1 1
1 1 1 1 1
-0.8
-0.6
—
1 * °
° ° °^
J^-^^"^^ y -
A""^
° < o" o ESTIMATED AVERAGE
-0.4
—
^ \
' ° I r ^. CORRELATION COEFFICIENT
-0.2
"
\,
°
Ll_
CI3
—
" 1
Ll_l
0.2
1_1_J
0.4
—
0,6
—
ESTIMATED I
° US. DATA
« GERMAN DATA
0.8
MEASUREMENT
ERROR 1
1.0
1 1
1 1 1 1 1
3
3
2
4 6
sib), dB 1
Figure 6.5. Correlation coefficients of seasonal cycles o/ Ng and Lb: night
{2000, 2200, 2400) versus the standard deviation of Lb-
the sensitivity of £" to a unit change in f{n). The intercept, a, is a func-
tion of path length, antenna heights, and terrain characteristics, and can
be derived from existing prediction procedures [9]. Comments on some
results obtained are given in the remainder of this section.
6.1.2. Parameters Derived From the N-Profile
It has long been recognized that the variations in field strength on
transhorizon paths are intimately connected with changes in the vertical
gradient of refractive index over the path. Although our knowledge of
the detailed fluctuations in refractive index in the troposphere is still in-
adequate for many requirements, it is nevertheless possible to relate,
234
TRANSHORIZON PARAMETERS
165
— I — I — I — I — I — I I I r
L(,- -0.230/^^*245.87
= 0.868 X
WJAS FM -PATH 12
TRANSMITTER: PITTSBURGH, PA.
RECEIVER: STATE COLLEGE, PA.
DISTANCE: 188.5 KILOMETERS
fi: 30.44 MILLIRADIANS
295 300 305 310 315 320 325 330 335 340 350
Ne
35 40 45 50 55 30 35 40 45 50
ANn,
165
AIMq.i
1
1 1
1
^h
= 0.529
/iNo 3*196.24^
X
^
170
'l^-an'
0.871 -^
^
X
-x8
175
^
1 1
Lfy=0.595A Nqs * 198.80
60 30
165
ai
^0.3
1
1
^b
-0.760 aNqj* 204.54^
X
^
170
-
rLi,-AN-0.866
"X
>^
x_
\
l^
XX
175
'
XX ^
X ^,jy^
y"^ X
1
1
ANo.5
1
L^=0.668aNio
1
*20l.32
X
'■li,-an= 0.873 ^
l^y"^-
X
8>
-^r X
XX X
-
^y»*
1
1
Figure 6.6. Monthly median basic transmission loss versus Ns or ANh /or a path from
Pittsburgh, Pa., to Stale College, Pa.
PARAMETERS DERIVED FROM THE iV-PROFILE 235
statistically, changes in signal level and functions of a parameter derived
from routine surface and upper-air measurements of pressure, tempera-
ture, and humidity.
Following the work of Pickard and Stetson [10], N , and AA'' have been
the subject of detailed studies by several workers [1, 2, 11, 12, 13, 14].
Only the more important conclusions are summarized here. Values of the
correlation coefficient, r, relating monthly median values of either A^^ or
AA'' and field strength, derived from a number of paths in diverse climatic
conditions [2], range from 0.4 to 0.95 with a median value of about 0.7.
An analysis has also been made of the results obtained by using (a) values
of AA'' obtained from the surface readings and at heights other than 1 km,
and (b) values of AN between different levels on the profile up to a height
of 3 km. A comparison of measured field strength at frequencies near
100 Mc/s on 20 paths, 130 to 446 km long, located in various part of the
United States, yields the following result: the use of N s gives as good a
correlation as any of the AA^ values, due to the high correlation between
the surface value and these differences. The values of r relating monthly
median values of A''^ and AA'' (the decrease in the first kilometer) have
been obtained for the United States [15], France, [16], Germany, [17] and
the British Isles [18]. They range from 0.60 to 0.93. Moreover, the data
are consistent with the assumption of a reference atmosphere in which A'^
decreases exponentially with height [15]. These results lead at once to
a consideration of the value of N s in predicting the geographical variation
of monthly median field strength. This question is considered in more
detail later in this section.
It has been shown that, where only past radio or meteorological results
are available, one obtains at least as good a prediction of the diurnal and
seasonal variations of field strength from long-term meteorological data as
from relatively short-term (say 1 year's) radio data. The annual cycle
may be represented by a single regression coefficient of 0.18 dB/A^'-unit
for either night or day; however, the regression coefficients for the diurnal
cycles lie between 0.2 and 1.1 dB/N-umt and vary with distance and
season, being greatest for paths between 175 and 200 km long and for the
winter months. The possibility of predicting the variation of hourly
median values of field strength within any given month by combining the
.seasonal and diurnal correlations has also been discussed [14]. It is recog-
nized that the development of a prediction procedure based on this ap-
proach must account for the complex sensitivity of field strength, E, to
changes in A'^s; this requirement in turn leads to considerations of season,
climatic region, distance and frequency. The results for the distance
dependence of the E-N s regression coefficient are, of course, intimately
connected with the propagation mechanism. In particular, in the case
of VHF paths about 200 km long there will usually be components in the
received field due to diffraction around the earth's .surface and scattering
236 TRANSHORIZON PARAMETERS
from randomly dispersed eddies; on occasions there will also be a semi-
coherent field arising from partial reflection at elevated stable layers.
Some aspects of this complex situation are discussed in section 6.2.
To sum up other important conclusions reached by the authors men-
tioned above, the correlation between E and N s
(1) increases with increasing variation of E or N s,
(2) is greater for seasonal cycles of night-time values of the variables
than for the midday values, and
(3) is greater for summer diurnal cycles than for winter ones.
Conclusions (1) and (2) are particularly important if we try to assess the
utility of A^s in prediction work in terms of signal data from areas in which
the variation of E is small (e.g., Western Europe), or in terms of data for
afternoon periods only.
Another parameter, closely related to AN, is the "equivalent gradient,"
Qe, proposed by Misme [19]. This is defined as that linear decrease of n
with height which produces the same amount of bending as the actual
inhomogeneous atmosphere over a given transmission path. The prob-
lem is illustrated in figure 6.7, where the dotted line represents the actual
ray path between T and P above an earth of radius a. It is required to
determine the curvature, p, of the circular, full-line path which is tangen-
tial to the real path at P and which corresponds to a constant value of
dn/dh in a fictitious atmosphere. Boithias and Misme [20] have described
a graphical method for calculating Qe ( = l/p), for example, when P is
located in the center of the common volume of the antenna beams.
Monthly median values of ^^ are obtained from the corresponding monthly
median values of dn/dh, and Misme [19] has given tables of Qe for various
path lengths for different months.
It is evident that Qe, like AA^", expresses the amount of refraction pro-
duced by the atmosphere and one might expect a high correlation between
the two parameters. This is known to be the case in some results quoted
by Misme [19] and which are illustrated in figure 6.8. Here the monthly
median values of AN and Qe (for a 300 km path) are compared, together
with the variation in N s, for Leopoldville in the Congo area. The corre-
lation between Qe and AA^ is high, but the maximum values in Ns in
December and January are accompanied by a local minimum in the
values of Qe and AA^. These results, and similar ones from Dakar (W.
Africa), have been quoted in support of arguments that N s is of limited
value in predicting seasonal and geographical variations in field strength
[19]. It is essential, Misme claims, to investigate the nature of the
N ,-AN correlation in separate climatic areas; the correlation seems poor
for some equatorial climates, probably because of the presence of semi-
permanent elevated layers. In these conditions, Alisme feels that an
exponential reference atmosphere and a correlation between N s and AA^
are not to be expected.
PARAMETERS DERIVED FROM THE A^-PROFILE
237
E
.3^
Figure 6.7. Geometry of the equivalent gradient.
60 . 1 370
50
g 40
30
AN
ge
350
330
J A S N D J F M A M J
MONTH (JULY-JUNE)
Figure 6.8. Comparison of monthly medians of ge, AN, and Na
(Leopoldville, Congo)
310
238 TRANSHORIZON PARAMETERS
While admitting the importance of an explanation of the results dis-
cussed above, it is difficult at present to accept the argument that Qe
should be used instead of AA^. Indeed, it has been shown [19] that for
a 471-Mc/s link, 160 km long, located in the Sahara region, there is a
difference of 20 dB between the signal levels exceeded for 99 percent of
the time during daylight hours in January and June, whereas the January
to June variation in Qe is less than 2A^/km. This result indicates the in-
fluence of atmospheric stability on signal strength. We defer, therefore,
further consideration of Qe until later in the chapter merely noting at this
stage that its derivation requires more detailed calculation (i.e., ray
tracing) than in the case of AA^, without any appreciable benefit.
6.1.3. Parameters Involving Thermal Stability
The role of stable elevated layers in tropospheric propagation has been
discussed by a number of scientists; for example, by Saxton [21], Friis,
Crawford, and Hogg [22], and in work done by the French [23] in an
important series of papers. These contributions have stimulated interest
in the development of radio-meteorological parameters which depend, in
part at least, on thermal stability, and this section considers some of the
characteristics of these parameters.
6.1.4. Composite Parameter
Misme [19] has discussed the Sahara results mentioned above in rela-
tion to the theoretical work of Voge [23]. In considering the effect of
elevated layers, Voge introduces a parameter, t], defined by:
area of one or several layers at a given height
V — : : r~.
the horizontal area, at the same height, visible from
receiver and transmitter
The value rj is an increasing function of the atmospheric stability measured
between two levels in the atmosphere. Stability may be defined as the
work, ATF, required to raise a unit mass of air from one level to another.
We have, therefore:
7] tends to as A IF tends to 0.
■q tends to 1 when AW is large.
The values of AW (in joules per gram) for Aoulef (Sahara) are given
by Misme for the January and June months in which measurements were
made over the 160 km path at 471 Mc/s. The relevant data are sum-
marized in table 6.1.
COMPOSITE PARAMETER 239
Table 6.1. Influence of thermal stability on signal level
(d = 160 km: f = 471 Mc/s: North Sahara Region)
Jan.
June
(/. (-N/km)
34.0
32.5
AW for 0. 7 to 2 km -
0.17
0. 0024
Relative signal level, 99 percent value, (dB)
20.0
0.0
The dominant propagation mechanisms in January and June are
thought to be "diffuse reflection" and "scattering," respectively, and on
this basis Misme uses the equations given by Voge to calculate the ex-
pected difference in the January and June signal levels. With various
assumptions regarding the properties of stable and turbulent layers, a
value of 19 (±5) dB is obtained for the ratio of predicted field strengths in
January and June, in good agreement with the measured value exceeded
for 99 percent of the time. This analysis is facilitated by the fact that the
two selected months are characteristic of well-defined climatic situations
in the Sahara region. Nevertheless, the results indicate that a parameter
which combines the concepts of the equivalent gradient and thermal
stability may be of general application. Misme [19] has therefore sug-
gested a parameter M of the form:
M = a ige - iO -\- b [AW]") (6.2)
where a, h, and n are constants, Qe is the equivalent gradient, and AW is
the thermal stability defined above for a 1 km height interval. With
a = 0.5 dB/N/km, M provides an estimate of the variation in field
strength, E, in decibels, caused by changes in equivalent gradient and
stability. It remains to define E in a "standard" atmosphere by selecting
a mean value of stability and a suitable value for the term h.
This composite parameter is thought by Misme to be more representa-
tive of the real atmosphere (especially in tropical areas) than A^^ and AA^,
and it is certainly of great potential value. However, only fragmentary
radio data are available for purposes of comparison in the references
quoted, and here again the precise value of the parameter in prediction
work can only be determined by a more comprehensive study.
6.1.5. Potential Refractive Index (or Modulus), K
This parameter is defined as the value of A'^ = (n— 1)10*^ which an air
mass would have if brought adiabatically to a standard pressure, say
lOOOmbar, assuming a constant humidity mixing ratio. Examples of the
use of K in radio meteorology have been given by Katz [24], and by Jehn
240 TRANSHORIZON PARAMETERS
[25]. It has also been applied by Flavell and Lane [6] in studies of the
effect of anticyelonic subsidence on tropospheric propagation. Values of
K can be derived very rapidly from upper-air data and, since plots of K
against height or pressure do not exhibit the large systematic decrease of
A^ with height in the conventional N{h) profile, the structure and motion
of meteorological features are clarified considerably. (Below the conden-
sation level, a lapse rate of — 20A''/km = dK/dh = 0.) In this respect,
K is superficially similar to the A unit [26] derived from the exponential
reference atmosphere. However, no quantitative results are yet avail-
able with this parameter and, furthermore, the method of deriving K
assumes a dry adiabatic lapse rate and a constant humidity mixing ratio.
The presence of a condensation level in the actual atmosphere is therefore
neglected. In addition, accurate values of A'' can only be derived directly
from the K profile in certain restricted conditions. Pending further
studies in this direction therefore, K and dK/dh remain more suitable for
qualitative synoptic studies than for quantitative predictions of field
strength variations. However, it is of interest to note that a close con-
nection exists between K and the composite parameter M discussed above.
Misme [27] has shown that the change of K in a given height interval,
AK/Ah, is of the form:
AK/Ah = ki AN/ Ah - k2AW/Ah (6.3)
where ki and ko are constants, and AW is a measure of thermal stability
as defined earlier.
6.1.6. Vertical Motion of the Atmosphere
The influence of stability has also been discussed by Moler and Arvola
[7], and Moler and Holden [8]. These authors assume that the average
lapse rate, dn/dh and the magnitude of local irregularities on the profile,
are primarily determined by changes in vertical velocity. Local centers
of convergence (low pressure cells) produce updrafts which result in con-
siderable mixing and the dissipation of any stable layer structure. Hori-
zontal divergence from local high-pressure centers create temperature
inversions and associated layer-type discontinuities in the n-profile.
These latter features are most pronounced in conditions of anticyelonic
subsidence.
The direction and relative magnitude of the vertical component of
wind velocity can be estimated by techniques outlined by ]\Ioler and
Holden, and a correlation between hourly median field strength and calcu-
lated values of vertical velocity has been noted by these authors. These
fundamental studies represent an important attempt to explain signal
variations in terms of atmospheric motion, and a survey of available ex-
perimental evidence supports the basic assumptions in this approach. It
VERTICAL MOTION OF ATMOSPHERE 241
is particularly interesting to consider whether local centers of convergence
and divergence, ill-defined on Daily Weather Reports, can be distin-
guished by observations with microbarographs and whether the results
are correlated with signal characteristics. Some results of such an investi-
gation, forming part of a more general study of VHF propagation in the
United Kingdom, are shown in figure 6.9. The signal records obtained
at a frequency of 186 Mc/s on a 140 km path are shown for 12 September
1959 together with the pressure variations recorded at transmitting and
receiving sites. The steady high signal at 0900 hours begins to fall as the
pressure at the transmitter begins to decrease, and at about 1030 hours
the median level and fading rate change abruptly as the pressure at the
midpoint falls. Between 1200 and 1700 hours the pressure over the whole
path is falling, and the signal characteristics remain essentially uniform.
However, a significant increase in median level and a reduction in fading
rate are evident at 1700 hours as the pressure begins to increase at the
transmitter. By about 2300 hours the pressure at the center of the path
has reached a steady value and the signal level is again high with negligi-
ble fading. The total barometric variation between 0900 and 2200 hours
is only 2 mbar.
There is cbviously a fruitful field of radio-meteorological study sug-
gested by the work outlined above, but we have to conclude that the
results obtained, while extremely valuable in clarifying a qualitative
relationship between signal strength and vertical motion (or stability),
are not immediately applicable in the problem of predicting field strength
variation.
6.1.7. Discussion of the Parameters
The above review emphasizes the need for a critical inspection of
representative data, radio and meteorological, with the objective of com-
paring the merits of the several parameters and explaining, if possible,
some of their relative merits and limitations. Some relevant points have
been mentioned already, and these issues are developed m more detail in
the following discussion.
It is important to bear in mind that in the particular problem of predict-
ing field strength changes we are concerned with a statistical relationship
between two quantities, the median signal level and a radiometeoro-
logical parameter. It may be advantageous, of course, to investigate
the merits of special parameters developed for particular climatic
areas; for example, the value of the gradient, dn/dh, in a semipermanent
elevated layer in the trade winds area. Nevertheless, such a parameter
must also be statistically "reliable" if it is to be applied in prediction
analysis.
242
TRANSHORIZON PARAMETERS
IS SEPTEMBER I9S9
2ioC e.M.T.
1019
e 1018 ^^- — -^^
D
-Q
E
UJ
(T
Z)
V)
en
UJ
(r
a.
lOIT
1016
1015
t\
\
\
/^
s
/ /
/ /
/ /
/
/
CLEAR SKY
1
CLOUD
i
1 1
/
CLEAR SKY
1
1 1
0600
1200
TIME (GMT)
1800
12 SEPT 1959
2400
Figure 6.9. Signal records and pressure variations at transmitter, T, and receiver, R,
on a 140 km path; f = 186 Mc/s.
COMPARISON OF SOME PARAMETERS 243
6.1.8. Comparison of Some Parameters
It is probably fair to say that, at present, the greatest interest is con-
centrated on the relative merits of N s, AN, and Qe, for these quantities
have received more detailed attention than any others. (It might be
argued that, since the initial gradient of refractive index with height has a
strong influence on refraction, this quantity should be highly correlated
with signal level. However, the errors of measuring the initial gradient
by radiosonde techniques effectively mask the correlation with the signal.)
In a comparison of N s, AN, and Qe, the monthly median values of trans-
mission loss, L, were determined for the hours of radiosonde ascents for
20 radio paths in various parts of the United States.
The equivalent gradient, Qe, was then calculated using standard ray-
tracing techniques [15] assuming:
(a) the actual antenna heights,
(b) a smooth earth with the sea-level radius,
(c) horizontal stratification of A'^ with a vertical distribution the same
as that of the monthly mean.
Standard statistical methods were then employed to determine the cor-
relation between monthly median values of (a) E and Qe, (b) E and N s,
and (c) E and AA'^ (surface to 1 km height). The average values of these
correlation coefficients for the 20 paths are given in table 6.2. The paths
studied in this comparison were the same as those listed by Bean and
Cahoon [2] in their analysis of N ^ —AN correlation. The data used were
representative of climatic conditions ranging from those of New England
and the Great Lakes area to Texas and the Pacific coast. This investi-
gation, at least, would seem to justify existing prediction procedures
based on A^^ and AA^. It is valuable, however, to consider some of the
results for specific paths in more detail, since they illustrate some features
of interest directly relevant to the arguments concerning the value of
N s and AA'^.
Table 6.2. Average correlations, r(z), of refraction variables with field strength
(20 paths, 130 to 446 km long,/=92 to 106 Mc/s)
Variable
ge
N,
AiV
r(z)
0.59
0.70
0.71
6.1.9. Some Exceptions and Anomalies
In so complex a matter as field-strength prediction, it would be un-
reasonable to expect any radio-meteorological parameter to be of world-
wide application to a uniform degree of reliability, and some conditions
have already been quoted in which N s may not be a reliable parameter.
The significance of these examples is really the issue on which views
244 TRANSHORIZON PARAMETERS
diverge most at the present time. Unfortunately, adequate radio data
are not available for several areas of interest (e.g., equatorial Africa) and
in these cases we can only propose tentative explanations based on a
critical examination of existing results, for similar but not identical con-
ditions.
An examination of available data, in published and unpublished reports,
has shown two particular examples which deserve further study; these
are (1) climatic conditions in which stable elevated layers are persistent
during certain seasons of the year, and (2) equable climates in which the
annual range of N s, AN, and field strength is relatively low. Attention
has already been drawn to an example in the first category; namely, the
path from San Diego to Santa Ana, Calif. [2]. The well-defined coastal
inversion in this area occurs at a height of about 0.7 km, and the asso-
ciated stable layer has a strong influence on radio field strength. Here
the correlation between field strength and A^ gradient for heights up to
0.7 km above the surface is small and negative, (i.e., opposite to the general
trend). The correlations with N s, however, and with N differences to
heights above the base of the inversion, are about 0.8. This result sug-
gests an explanation of some of the results, already mentioned, for Dakar
and Leopoldville. Consider the profile of figure 6.10 typical of ascents
made through daytime inversions in equatorial and Mediterranean areas.
(For comparison purposes, an exponential reference atmosphere is also
shown.)
At Dakar, for example, an elevated layer is observed in 40 to 50 percent
of the daytime soundings during August, frequently with the base of the
inversion above 1 km height. In these conditions, at least, the high N s
values measured in August are not accompanied by high values of AN
(0 to 1 km), for the lapse rate, dN/dh, below the inversion is generally less
steep than would be expected for the given value of N s- It would be
valuable, therefore, to examine in more detail the distribution of the
height of the layer, particularly in the summer months. If the data given
by Misme contain an appreciable fraction of such profiles, then the poor
Ns — AN correlation he discusses is not surprising. The correlation be-
tween N s and signal level, however, could still be significant, as in the
San Diego-Santa Ana link [2]. Further radio data are obviously required
for areas in which elevated inversions are persistent.
In many temperate regions, a somewhat different situation exists. In
western Europe, for example, the annual range of monthly median values
of A" s is 10 to 20 N units; similarly, the variation in the monthly median
values of field strength is also small and frequently lies within the esti-
mated measurement error. Figure 6.11 shows some results which illus-
trate these features. Monthly median values of relative field strength
are shown for the months January-November, 1959, for a 300-km path at
a frequency of 174 Mc/s; the terminals being located at Lille (northern
SOME EXCEPTIONS AND ANOMALIES
245
£
X
o
LiJ
X
Figure. 6.10. Typical profile through an inversion layer {equatorial and Mediterranean
areas.) _^
20 y^
\— X.
CiTX:
\
X Ns
o RELATIVE FIELD STRENGTH
J L
J L
15
X
I-
o
UJ
or
CO
10 Q
_i
LU
5 ^
<
_j
n UJ
JFHAMJJASOND
MONTH OF YEAR, 1959
Figure 6.1 1 . Variation of monthly median values of Ng and field strength for a 300 km
path from Lille, France, to Reading, England.
246 TRANSHORIZON PARAMETERS
France) and Reading (England). The variations in N s obtained from the
Crawley radiosonde station (close to the midpoint of the path) are also
shown. Here N s has a range of ± 10 A^ units, and the signal level a range
of no more than ±4.5 dB. In fact, from March to September the total
variation in observed monthly median field strength is only ±1.5 dB.
The measurements of field strength in these experiments were estimated
to be subject to possible errors of about ±3 dB. Consequently, one
would not be justified in assessing the value of A^^s as a radio-meteorological
parameter on the basis of the correlation coefficient (about 0.25) calculated
from the two curves in figure 6.11. During the period studied in this
work (0900-2300 UT daily, January-November 1959) the highest signals
were observed during anticyclonic winter weather, with elevated inversion
layers at a height of about 0.6 km. Extended stable layers at this height
were rarely seen on the sonde ascents during the summer months. This
result may partly explain the fact that the seasonal variation in Ns
(highest values in the summer months) is not accompanied on the Lille-
Reading path by a corresponding variation in monthly median field
strength.
6.1.10. Conclusions
A review of available data shows that no radio-meteorological param-
eter has yet been proved to be superior to A^s or AA^ for general application
in the prediction of field strength distributions. The value of these
parameters has been established by studies of many paths in diverse
climatic conditions. However, it should be noted that much of the
radio data has been obtained at frequencies near 100 Mc/s, and there is
a clear need for further analysis of the several parameters in conjunction
with field strength measurements at higher frequencies.
Pending more detailed results, it does not seem likely that the equiva-
lent gradient, Qe, affords any significant advantage over AA'', especially
in view of the many calculations required in its derivation. Some data
from selected areas (Congo, Sahara, west Africa) suggest that A''^ and AA''
may have limited value in these regions; however, the present lack of
adequate radio data precludes any definite conclusions. These examples,
and allied work on vertical motion and thermal stability, emphasize the
importance of a parameter (such as that suggested by Misme) which takes
account of the influence of elevated layers. This approach would prob-
ably prove fruitful not only in equatorial areas but also in temperate
regions where the annual range of N ^ and AN is small.
Analysis of some results obtained at a frequency of 100 Mc/s has shown
that it has proved feasible to provide at least as good a prediction of the
annual cycle of field strength variations from long-term meteorological
data as from relatively short-term radio data.
INTRODUCTION 247
6.2. An Analysis of VHF Field Strength Variations
and Refractive Index Profiles
6.2.1. Introduction
It is evident from the discussion in section 6.1 that the further develop-
ment of radio-meteorological parameters would be assisted by a better
understanding of the propagation mechanism on non-optical paths. In
particular, the influence of thermal stability on signal level, fading rate
and wavelength dependence is an important topic requiring further study.
The effect of varying meteorological conditions on signal characteristics
is especially marked in the case of paths of "intermediate" length, with
terminals just beyond the radio horizon. As noted in section 6.1 on such
a path the residual scattered field in the absence of stable layers or surface
ducts will often be comparable with the weak diffracted field. Further-
more, at frequencies up to, say 300 Mc/s (X > Im) relatively strong
fields will frequently be observed in conditions favorable to the production
of temperature inversions in the first 2 km or so above the earth's surface.
It is the purpose of this section to discuss some aspects of radiometeor-
ology relevant to this situation, especially the field, strength distribution
observed on a 200 km path at frequencies between 72 and 180 Mc/s.
The signal characteristics are analyzed in terms of a classification of
refractive index profiles, with the objective of clarifying the relative im-
portance of different propagation mechanisms and their influence on the
measured field strength distribution.
Table 6.3 lists characteristic profile types, the assumed mechanism
associated with each type, and typical meteorological conditions. Se-
lected references are given for each category, and special mention may be
made here of recent work [28] in France which, to a large extent, unifies
and extends earlier analyses based on the separate concepts of "reflection"
and "scattering."
6.2.2. Radio and Meteorological Data
The analysis to follow is limited to paths between Chicago and Urbana,
111. (fig. 6.12), since several years of radio data were available for four
separate wavelengths between 1.67 and 4.18 m. Moreover, two radio-
sonde stations are located on or near the path, a unique situation in radio-
meteorological investigations. Details of the radio paths are given in
table 6.4 (in which 6 is the total angle between the horizon rays from trans-
mitter and receiver on a 4/3-earth profile).
The meteorological data were obtained from the simultaneous radio-
sonde observations made by the Weather Bureau at Joliet and at
Chanute Air Force Base, near Rantoul, 111. The results used were those
from the significant levels reported whenever the temperature or humidity
departed by ±10 percent from predetermined values.
248
TRANSHORIZON PARAMETERS
Table 6.3. Refractive index profile classification, probable propagation mechanisms
and meteorological conditions
Profile
Assumed propagation
mechanism
Ref-
Meteorological
conditions
Unstratified (U), monotonic decrease
with height, gradient nowhere ex-
ceeds twice normal for that height.
Elevated Layer (EL), monotonic de-
crease with height with one or more
distinct layers with gradients at least
twice normal for that height.
Super-refractive (SR), same as EL but
the layer is ground-based.
Ducting (D), same as SR but the gra-
dient exceeds the earth's curvature,
1/a.
Scattering plus diffraction
Scattering plus diffraction
plus reflection.
Extended radio horizon pro-
ducing enhanced dif-
fracted and scattered
components.
Extension of radio horizon
to include the receiver.
[29]
[30]
[31]
[32]
[21]
[28]
[31]
[33]
Well-mixed atmosphere
due to thermal convec-
tion and, or wind shear
Layer formed by subsi-
dence inversion or lift-
ing of radiation inver-
sion.
Radiation inversion
formed during the
night or rapid evapo-
ration from soil after
rain.
Same as SR.
Table 6.4. Chicago-to-Urbana radio path characteristics
Station
Distance
d
X
/
Period of record
WBKB-TV _...
WNBQ-TV
WMBI-FM
WENR-TV
km
203.1
202.9
202.7
202.9
mrad
16.3
16.7
16.1
16.4
COT
4.18
3.67
3.15
1.67
Mc/s
71.75
81.75
95.50
179. 75
5/51- 5/53
10/50-10/51
7/50- 6/52
7/51- 6/53
6.2.3. Classification of Radio Field Strengths
by Profile Types
The RAOB significant level data were converted to refractive index by
use of (1.20). The gradient of A^ was then determined between the re-
ported significant levels of each profile and examined as to whether the
gradients fell into the category of linear, subrefractive, or superrefractive
depending upon the criteria set down in table 6.5, wherein superrefractive
is approximately twice normal and subrefractive has a positive gradient.
Simultaneous observations of similar profile types at Rantoul and Joliet
were necessary for entry as a distinct profile occurrence. If a super-
refractive layer occurred above the crossover heights of the radio horizon
tangent rays from both transmitter and receiver then it was classified as
an elevated layer provided the reported layer heights were within 1 km
of one another at both radiosonde stations.
Table 6.5. N gradient classification of profile types in N-units/km
p
h
Subrefractive
Unstratified
Superrefractive
mb
1000-850
850-700
700-600
600-500
500-400
km
-1.46
1.46-3.01
3. 01-4. 20
4. 20-5. 57
5. 57-7. 18
-(dn/dhXQ
-{dn/dhXO
-(.dn/dhXO
-(dn/dhXO
-(dn/dhXO
20<-(dH/dft)<60
20<-(dn/d/i)<50
15<-(d;i/dft)<40
10<-(dn/d/i)<30
10<-(dn/d/i)<25
100<- (d«/dft)
80<- (dn/dft)
■JQ<-(dn/dh)
50<- (dn/dh)
40< -(dn/dft)
RADIO FIELD STRENGTHS BY PROFILE TYPES
249
Elevated layers below the crossover height were classified as ground-
based superrefractive layers. Elevated layers below the crossover height
at one weather station and above that height at the other were classified
as tilted elevated layers.
After these characteristic profiles were isolated, the median field
strengths for the 3-hourly periods centered upon the radiosonde observa-
tion time were arranged into cumulative probability distributions for
each profile type. The results are shown in figure 6.13. (There were
relatively few examples of subrefractive profiles and no distributions for
this category are given.)
J.. OHIO ,
1 \.
<^^-J.
MISSOURI V
KENTUCKY
.r^
Figure 6.12. Location of radio path and radiosonde stations used in this study.
250 TRANSHORIZON PARAMETERS
Generally, the unst ratified -samples have the lowest overall field
strengths throughout the entire distribution range. The presence of any
layer (elevated or ground-based) tends to increase the field strength by
10-25 dB at any percentage level of the distribution. (The exception to
this observation, WXBQ-TV, is probably explained by the fact that the
observations were limited to six winter months, rather than the 2-year
period of the other stations.) Tilted elevated layers appear to produce
the greatest enchancement of signal strength, probably as a result of
focusing effects due to the layer tilt.
6.2.4. Prediction of Field Strength for
Unstratified Conditions
The field strengths recorded during the times when the radiosonde
ascents at both Joliet and Rantoul indicated nonstratified conditions were
compared with the values predicted by Norton, Rice, and others [31] for
the case of diffracted plus scattered radio waves. This particular pre-
diction process is adjustable for the average refractive conditions over the
path in that it adjusts the effective earth's radius factor to the initial
gradient of A^ for the calculation of diffracted field strengths. One also
needs the angular separation of the radio horizon rays at their intersection
near midpath. The average initial gradient of A^ was obtained for each
instance of unstratified profile by .simply averaging the initial gradients
from Rantoul and Joliet, while the angular separation was obtained by
determining the amount of radio ray refraction expected over each particu-
lar path in atmospheres of exponential decrease with height that closely
match the observed A^ conditions.
Figure 6.14 illustrated a comparison of the predicted and observed field
strengths. For WNBQ and WENR there is approximate agreement be-
tween the two sets of data. However, the predicted values for WBKB
and WMBI are approximately 10 to 12 dB higher than the observed
values.
This tendency to predict field in excess of the measured values suggests
that the empirical data, on which the predictions are based may include
meteorological conditions with some degree of stratification in the first
2 km or so, even though most of the empirical data refer to afternoon
hours in winter. It will be shown that elevated layers of moderate size
(say a few kilometers in horizontal dimensions), that may exist undetected
by the radiosonde, could produce field strengths on the Illinois paths com-
parable with the median values for "unstratified" conditions shown in
figure 6.13. Furthermore, the limits placed on the profile gradients
specifying unstratified conditions in table 6.5 are such that some layer
type profiles may be included in the unstratified category. Consequently,
it is important to study in more detail the properties of the elevated layer,
ELEVATED LAYERS ON THE ILLINOIS PATH
251
WBKB -TV
PATH 206
--
\
\
IINSTRATIFIEO 1188)
GROUND BASED (53) _
-
t
N
- GB +EL (23)
- ELEVATED LAYER (339)
■.
_N
■^^
^^
~^N
X
-
-
\
^Ot
-
-
\,
\
■•■■X_^,
\-
-
\
\
\
N
-
0.5 I 5 10 30 50 70 90 95 99 99.5 99.9
PERCENT OF TIME VALUE EXCEEDS ORDINATE
WNBQ -TV
H 219
1
^
■~N
UNSTRATIFIED (259)
GROUND BASED (31)
^\
GB+ EL (10)
-
1
\
\
ELEVATED LAYER (73)
■■•-...
N
~
V
-
'"\,
V.
-
~
N
-
-
\
^"^-^
-
'^^
■~-
— .
-
^
-
-
0.1 0.5 I 5 10 30 50 70 90 95 99 915 99.9
PERCENT OF TIME VALUE EXCEEDS ORDINATE
Figure 6.13. Distribution of hourly median field strengths with different radiosonde
profile conditions.
not only as a feature occasionally producing high field strengths, but also
as a mechanism which, in less intense form, partly determines the strength
of the weaker fields observed for large percentages of the time.
6.2.5. The Effect of Elevated Layers on the Illinois Path
The influence of elevated layers of VHF transmission beyond the
horizon has been studied by several workers [21, 22, 28, 34, 35, 36, 37].
However, few investigations have contained any detailed comparisons of
theory and experimental results. The following analysis presents such
a comparison, using simple models of the elevated layer for the four
Illinois paths.
252
TRANSHORIZON PARAMETERS
6.2.6. Elevated Layers at Temperature Inversions
Recent radar and refractometer investigations of tropospheric structure
have shown that elevated layers in the refractive index distribution are
frequently observed in the stable air of temperature inversions [38].
A typical value of layer thickness is 100 m, with horizontal dimensions
of tens of kilometers. On occasions, extended layers no more than 10 m
in thickness have been detected by refractometer soundings. In the
present discussion we attempt to evaluate the reflection coefficient of
these elevated layers. We may express the modulus of the reflection
coefficient \p\, for a wave incident at a glancing angle a on a layer of
thickness h, and horizontal extent, x, in the form:
An
f(a, h, x)
(6.4)
where f{a,h,x) is the ratio of the reflection coefficient of the model to
that of the infinitely sharp case (i.e., the Fresnel discontinuity value,
An/2a2). This function has been evaluated for several layer profiles [28],
-
' i
WBKB-
PATH
206
-
-
L
NSTRi
TIFIED
FIELC
STRE
NGTHS
■
-
/
1 .
• f
•-
/
_•
1
t
,..s!
i •'
Yy. .
-
:
I . I
r ~*
.;^.:::
^.
-
-
1
[
/^
-PREDICTED VALUE -
1 1
,
^
24
o
u.
H
o
5 12
'
'
/
'
-
WNBO - PA
1
TH 21=
/
-
/
-
^
/
.
, .
-
•
/
).-.
-
-
**h:
*
-
/
i , i
'
-
' M ■ i
'
! '
-
WUE
l-PA
TH 57
■
,
/
- -
•
■ .
"..
. • •
/
••
I._V
r A
V*
/
-
■V. t
-/
-
_•
•
"if- .4.« /
, /^
-16 -12 -8 -4 4 8 12 16 "0 4 8 12 16 20 24 28 32 36 40
OBSERVED FIELD OBSERVED FIELD
28 r
24
_i
LlJ
il 16-
Q
UJ
h 12-
-12 -8 -4 4 8 12 16 20 24 16 -12 -8 -4 4 8 12 16 20 24
OBSERVED FIELD OBSERVED FIELD
-
' 1 ' ! ' i '
WENR-PATH 210
1
-
/
-
/
-
• •
J-
/
-
! 1
/
...
-
•
•
!"
ry-
;•^
-
■ /
-
/
• •
•
Figure 6.14. Com-parison of observed radio field strengths and values predicted for
wintertime afternoon hours.
ELEVATED LAYERS AT TEMPERATURE INVERSIONS 253
and preliminary calculations based on this work were made to determine
the most suitable model in the present application. It was evident from
these calculations that a simple linear profile would yield the best agree-
ment with the measured data, and this model was therefore adopted in
the subsequent analysis.
Consider the layer profile shown in figure 6.15, i.e., a linear decrease of
An over a height interval h, with transition regions of height d. This
model and others have been discussed by several authors, but the most
detailed treatment is that of Brekhovskikh [39]. His analysis shows that
for this linear model:
I p I = An • X/Sirh sin^ a
~ An • X/Sirha^. (6.5)
This equation is valid if:
(a) An • \ <K rha^
and (b) 4ad « X.
In the present problem, with values of X of 1.7 to 4.2 m, An '^ 10"^,
a ^ 0.02 condition (a) is satisfied for layer thicknesses greater than about
20 m. In addition, condition (b) is fulfilled for the stated conditions if the
thickness of the transition region is less than a few meters. These condi-
tions do not seem inconsistent with available refractometer data on
elevated layers, but a rigorous justification of the model is impossible at
the present time. In any case, there is almost certainly no unique profile
representative of all elevated layers. We assume here, therefore, the
linear profile of figure 6.15 merely as a simple analytical model. It may
be noted here that the value of | p | given by (6.5) agrees with that quoted
by du Castel [40] but is half the value obtained in an earlier analysis [28].
Equation 6.5 was used to calculate reflection coefficient of the layers
on each occasion on which these were observed in the sonde ascents. The
results, expressed in terms of a reflection loss are compared with the
measured values of field strength in figure 6.16. The general agreement
is satisfactory for the assumed model. As might be expected, there is
a considerable scatter in the data and two considerations are important in
assessing the significance of these results. These concern sonde response
and layer structure. The work of Wagner [41] on the response of radio-
sondes shows that, for an elevated inversion layer with An = 3 X 10~*,
h = 100 m, a sonde with a 10 sec time constant in the sensing elements,
and rising at 5 m/sec will give an indicated value for An of approximately
half the true value. The above procedure, using sonde data, therefore
underestimates the value of |p| for an idealized infinite layer.
254
TRANSHORIZON PARAMETERS
X
(3
LJ
X
REFRACTIVE INDEX
Figure 6.15. Linear profile model of the n decrease across an elevated layer.
On the other hand, the analysis assumes a smooth layer extending
horizontally at least over a distance x equal to the first Fresnel zone.
We have:
X = ■\/2a\/a
(6.6)
where 2a = path length = 203 km for the Illinois paths. Hence x is of
the order of a few tens of kilometers. In addition, we have assumed that
the layer is horizontal and smooth over a distance x, thus neglecting con-
vergence. If we adopt the Rayleigh criterion, the height of the surface
INFLUENCE OF SMALL LAYERS 255
irregularities on the layer, A/i say, must not exceed ±X/8a for the layer
to be considered smooth; i.e.,
Ah < ±7 m (X = L67 m)
Ah < ±17 m (X = 4.18 m).
These values apply for a = 0.03 rad, corresponding to a layer height of
2.5 km; for lower layers Ah will be greater due to the decrease in a.
These conditions are not likely to be satisfied in all the examples studied
and the analysis therefore overestimates the value of lp| in this respect.
(Some discussion of this point has been given by Bauer and Meyer [42].)
The limitations of sonde soundings, and the effects of layer tilt and surface
irregularities therefore provide a partial explanation of the scatter of
points in figure 6.16. Further detailed measurements of layer structure
are obviously desirable.
6.2.7. The Influence of Small Layers
The above discussion has dealt with the particular case of extended
elevated layers such as are often associated with temperature inversions.
However, it seems quite possible that these layers are merely the more
extreme examples of anisotropic irregularities which are thought to be
prevalent in the troposphere. There is already some preliminary evidence
supporting this concept in the results of refractometer and radar soundings
and recent theoretical work [22, 28, 40] has developed this approach in
detail. The relationship of this work to earlier analyses in terms of a
"scattering" model is discussed in the references quoted and need not
concern us here. For our purpose it is sufficient to utilize the essential
features of the argument as the basis for a simple calculation.
It seems reasonable to assume that even in an atmosphere which sonde
data would lead us to classify as "well-mixed" there are often layer-type
irregularities. Detailed evidence on the spatial form and stability of this
type of layer or "feuillet" is so far lacking, but an inspection of some
refractometer results suggests that horizontal dimensions of a few kilo-
meters represent a realistic assumption. Such a layer might exist as a
separate entity for, say, several minutes (as compared with a period of
several hours for the extended layer in a stable inversion).
For the following analysis, let us consider two layers of horizontal
dimensions, x, of 2 and 10 km respectively with the following character-
istics:
An = 10-5
/i = 100 m
a = 0.01-0.03 rad (i.e., layer height of 0.4 to 2.5 km on the Illinois
paths).
256
TRANSHORIZON PARAMETERS
35
30
1 'WB
B-TV
/
L
NEA
R PR
OFIL
A
■'
1
/
20
15
10
\
•
/
.
/
y
•,
•
/
y
'•
'•
»,
/*
.
•• J
•
:
/
^
-f
RED
CTED
VAL
UE
/
.
A
-Lr REFLECTION LOSS IN dB
WMBI - FM
/
/
/
5
/
•
.V
••
-/•
•
y
•
•
\
'•/
-•.'
.
•••
'/-
.
.
.
/
•
/
/
•
50
1
1
9
T
5
3
10
-Lr REFLECTION LOSS IN dB
WNBQ-TV
/
/
■ /
/
•
/
/•
•
•
/
/
y
. t
/
/
•
/
r
-Lr REFLECTION LOSS IN dB
WE^
R-T
V
—
—
—
~
—
—
1
—
~
"
4
—
#
/
^
r
y
..
/
,
»,
r^
/•"•
/
y
..
y-
/*
•
•
/^
M
-Lr REFLECTION LOSS IN dB
Figure 6.16. Observed hourly median field strength versus reflection loss assuming the
linear model of figure 6.15.
For these conditions, the layers correspond to those of "intermediate"
size in the analysis of Friis, Crawford, and Hogg [22]. They are defined
by the inequality:
\/2a \/a > X > \/2a\
(6.7)
where 2a is the path length. In this case, the power received, Pr, from
an antenna of effective aperture, Ar, with a transmitter radiating a
power Pt from a antenna of effective aperture ^r is given by:
Pr/Pt = {AtAr X2aV)/(2XV).
(6.8)
INFLUENCE OF SMALL LAYERS 257
We can use this equation to calculate the corresponding field strength,
for the Illinois paths, in terms of lA'/m for 1 kW radiated from a half-
wave dipole. We have the following relations:
A (X/2 dipole) = 0.127 X^ (6.9)
p, (X/2 dipole) = ^2X2/300^2 (6.10)
where E is the field strength in volts/meter if Pr is in watts. From
(6.8), (6.9), and (6.10), we can calculate E for the two layers specified
above. The results obtained are shown in figure 6.17 for various layer
heights and the following models of reflection coefficient:
(a) \p\ = An • \/8ira^h
(b) IpI = An/2a\
Model (b) is the Fresnel discontinuity equation which gives the limiting
value of \p\ towards which all models tend as the layer thickness de-
creases. The curves in figure 6.17 show that the calculated field strength
depends considerably on the assumed n-profile. If | p | = An • X/Sira^h,
values of field strength comparable with the long-term median value may
be produced by layers of about 10 km in lateral dimensions in the height
range 0.5 to 1 km. If |p| = An/2a^, similar field strength may be pro-
duced by layers in this height range if the lateral dimension is of the order
of 2 km. The effect of the layer decreases with increasing height, but
even with layer heights of 3 km, the field strength is still 1 /LtV/m or
greater at both wavelengths for a 10 km layer with \p\ — An/2a'^. How-
ever, it should be pointed out that the assumed value of An = 10~* is
probably somewhat large for layers as high as 3 km. The results also
show that model (b) (i.e., p = An/2a:-) gives field strength values which
are higher at X = 1.67 m (/ = 179.75 Mc/s) than at X = 4.18 m (/ =71.75
Mc/s).
6.2.8. Conclusions
Any departure of refractive index structure from a smooth monotonic
decrease with height produces an increase in field strength on a 200-km
path in the frequency band 70 to 180 Mc/s. In the particular case
studied, elevated tilted layers result in signal enhancement of 10 to 25 dB
over the values for unstratified conditions, at all percentage levels. (The
importance of the tilted layer is possibly a consequence of the asymmetry
of the path, the transmitting antenna being 200 m above ground and the
receiving antenna 30 m.)
The predicted field strengths, for conditions classified as unstratified
in terms of sonde data, are in approximate agreement with observed
results, although the scatter of the points (plus the tendency to predict
258 TRANSHORIZON PARAMETERS
values in excess of the measured ones) point to the influence of anisotropic
layers or eddies of varying size and degrees of stability. This interpre-
tation is consistent with numerical calculations based on the properties of
"intermediate" size layers, suggested in the analysis of Friis, Crawford,
and Hogg [22].
Calculations of the field strength produced by extended stable layers,
using sonde data and a model profile with a linear lapse of n with height,
are in reasonable agreement with the experimental results. However,
there is probably no unique profile characteristic of elevated layers.
6.3. A New Turbulence Parameter
6.3.1. Introduction
We shall here attempt to unify past work utilizing the surface value
of the refractive index and the concept of atmospheric stability. To do
so we shall first review the concepts of atmospheric stability from the
viewpoint of temperature structure. In the process we shall show that
past efforts have neglected the important role of the conditions at the
earth's surface; then we shall extend this work to derive an analogous
expression for the radio refractive index.
6.3.2. The Concept of Thermal Stability
The stratification of the earth's atmosphere by and large reflects the
control of the earth's gravitational field. This average structure is,
however, constantly disturbed by the intrusion of thermal plumes of
heated air rising from the differentially heated surface of the earth, as
well as by the mechanical mixing produced by the passage of air currents.
It is customary to assume that parcels of air are forced upwards through
the normal stratified atmosphere without mixing with the environmental
air mass. Since the parcel is forced to rise, be it by convection or by
mechanical mixing, it is also assumed that it expands and cools without
exchange of heat, i.e., adiabatically. In such a process it can be shown
that the pressure, P, is given by
where T is the temperature (°K); g, the acceleration of gravity; R, the
gas constant for air; and, where a, the constant lapse of temperature with
height is
(6.12)
CONCEPT OF THERMAL STABILITY
259
en
2
1.0 1.5 2.0 2.5 3.0
LAYER HEIGHT (km)
1.5 2.0
LAYER HEIGHT (km)
2.5
3.0
Figure 6.17. Field strength produced by layers of horizontal dimensions 2 km and 10
km on the Chicago-Urbana paths for two model profiles.
260 TRANSHORIZON PARAMETERS
The zero subscripts in (6.11) refer to conditions at the earth's surface.
Equations (6.11) and (6.12) are taken to hold as long as saturation does
not occur. It is further assumed that the mixing ratio
e^eo-^ (6.13)
-to
is constant throughout the process. In (6.13) e is the partial pressure of
water vapor.
A parcel of air which is caused to rise will cool to the dew point if forced
aloft far enough. Further rising will lead to condensation. Conversely,
a parcel of air which sinks down will have work done upon it adiabatically
and will become warmer than it was in its original elevated position. A
parcel of air following a condensation curve has a value of a, of — 6°C/km
denoted at a*. In addition, as water vapor condenses out of the parcel,
the vapor pressure will decrease. We here assume that the vapor pressure
will follow that of the saturation vapor pressure curve, Csz, as given by:
esz = est exp [-^a*(z-l)], z > i, (6.14)
where i is the height of the lifting condensation level and ^ = 0.064 (°C)~^.
The value of ( has been found to be given [44] with sufficient accuracy for
practical applications by
^^0.125 (T-Trf)o (km) (6.15)
where the zero subscript indicates that the difference between the tem-
perature and the dewpoint, Td, need be evaluated only at the earth's
surface.
Consider now a parcel rising through some environmental distribution
of temperature. The dry adiabatic lapse of temperature with height is
greater than that observed on the average in the atmosphere (6 °C/km)
and thus the parcel becomes cooler, and more dense, than the environ-
mental air and will sink to its initial conditions with the removal of the
lifting force. If, on the contrary, anywhere in its trajectory it becomes
warmer, and thus less dense, it will become unstable and rise of its own
buoyancy through the environmental air. Past radio-meteorological
studies have taken the area between the adiabatic curves and the environ-
mental temperature distribution on a pressure ('^ height) versus tempera-
ture chart as a measure of atmospheric stability. On the general argu-
ment that the effects of atmospheric stability, or turbulence, influence
radio waves only through the radio refractive index, we shall extend the
above concepts to the radio refractive index.
CONCEPT OF THERMAL STABILITY 261
6.3.3. The Adiabatic Lapses of N
Introducing the dry adiabatic variations of P, T, and e into (1.20),
chapter 1, one obtains the dry adiabatic decrease of A'^, N d-
N, =N.[1- y-j |1 - -^^i^l (6.16)
which is vaHd for 2 < (. The distribution of A" d with z may be evaluated
by taking the logarithms of both sides of (6.16)
(6.17)
and expanding the logarithms on the righthand side of (6.17) with the
omission of second-order terms where appropriate for our applications
of 2 < 3 km:
In Ad~ln Ao - r^^ (6.18)
or
Arf~A^oexp i-Tdz) (6.19)
where
When evaluated for the standard conditions of To = 288 °K (15 °C) and
Po = 1013.2 mbar, r^ becomes
Yd = 0.08 (km)-i. (6.21)
It is sometimes more convenient to use the dry adiabatic scale height,
Hd-.
Hd = — = 12.5 km. (6.22)
1 d
Above the local condensation level, the expression for A becomes that of
the wet adiabatic, A^^,:
»..(..#•(»,(..«)--
+ I^, exp (I3a*z)j (6.23)
262 TRANSHORIZON PARAMETERS
where, for convenience,
n 77 A ^' H w 77.6 (4810) e.,
D( = 77.6 — and Wt = y~2
Applying the same approximations as in the derivation of (6.18), one
obtains
In iV^ = In A^, - r^z (6.24)
where
!„* fe+0«*^'
2c
For the standard conditions assumed above and eo corresponding to
60 percent relative humidity, H^ '^ 7.5 km. Again, the form of the wet
adiabatic lapse of A^ is given by
A^^ = A^, exp i-T^z), z> i. (6.26)
We shall now apply these results to the derivation of a refractive index
turbulence parameter.
6.3.4. The Turbulence Parameter, n
Analogously to the concept of thermal stability, we define n as the area
between the environmental N{z) curve and the appropriate adiabatic
decrease of A^. That is
n =
/ \N observed ~ N adiabatic j dz (6.27)
where the integration is arbitrarily taken from the surface to 3 km. This
then extends the integration well above the crossover height of the radio
horizon rays from transmitter and receiver normally encountered in
tropospheric propagation. The integral for n may be written
n = / Ndz - j No exp (-Tdz) dz
l>-
exp[-V^{z-i)]dz. (6.28)
TURBULENCE PARAMETER, II 263
Noting that
(6.28) then becomes, upon integration,
No
n = Ndz-^ir [exp (-Tdl)-l]
Jo Ad
+ ^»exp ^-r^^^) {exp [-r.(3-0]-l}. (6.29)
1 w
n is thus dependent upon both A'^ profile characteristics and initial condi-
tions. In fact, all the terms on the righthand side of (6.29) save
3
Ndz
are determined from the conditions at the earth's surface. One might ex-
pect, then, that 11 would constitute a useful radio-meteorological param-
eter since it includes both the time-proven parameter A^o, the concept of
atmospheric stability, and the integrated A''-profile characteristics. We
shall now apply this new parameter to the radio data presented in detail
in section 6.2. The values oi H a and Hy, utilized are those for standard
sea-level conditions; i.e., 12.5 km and 7.5 km, respectively.
6.3.5. Comparison of Radio-Meteorological Parameters
The various radio-meteorological parameters discussed in section 6.1
plus n as derived from the Rantoul and Joliet radiosondes were tested
for their relative utility by comparison with the radio data of WBKB-TV
and WEXR-TV (see table 6.4). Correlation coefficients between the
radio data (3-hourly medians) of each path and the various radio-
meteorological parameters were determined for each of the profile cate-
gories given in table 6.5 as well as for all available data. It was not
possible to fit all of the data into the various categories since these were
deliberately made very strict in order to explore the utility of each
parameter within each of the propagation mechanisms assumed for the
profile categories (table 6.3). Thus it is observed that the sum of the
individual categories does not equal all available data. The results of
these correlations are given in table 6.6. Before discussing these results
it is well to note that the crossover height of the tangent rays from each
end of the path were calculated by detailed ray tracing [15] for each radio-
sonde ascent. The values as determined from the Rantoul and Joliet
soundings were averaged. This average crossover height was then used
to determine the contribution to the values of 11 below the crossover
264 TRANSHORIZON PARAMETERS
height (n"), above the crossover height (n'), and Qe- As hsted in table
6.6, n represents the value of (6.27) for the height increment zero to 3 km.
Before proceeding, the method of calculating ge will be described. The
concept of the equivalent gradient is closely related to that of the effective
earth's radius. The curvature of the effective earth is given by:
1 1 , 1 dn
— = - -\- - -,T cos d
re a n ah
where r^ is the effective earth's radius; a is the radius of the earth; and
(1/n) idn/dh) cos d is the curvature of the radio ray.
The value of r^ is determined from the geometrical relationship
h ^-'^
where c?,, the ground distance to the crossover height, h^, has already been
determined by the ray-tracing procedures discussed in chapter 3. Thus
one obtains
dn \ _ 2hc 1
dhj ~ d;'
for 6 = and setting (cos d)/n c^ 1. In order to obtain the equivalent
gradient as a positive quantity in the more convenient refractivity units,
the following expression is used:
(i _ 2^Y
0- = Ka - t^r-
It is evident from table 6.6 that the results obtained with IT are com-
parable with those using 11', due supposedly to the crossover height being
on the order of 100 m for these paths. The value 11" appears to contribute
little to the final value of 11 and is poorly correlated with the radio data.
An interesting exception to this observation is the subrefractive case
where 11" yields a larger correlation coefficient than either n or U'. The
overall conclusion that one reaches from the data of table 6.6 is that the
most promising of the parameters considered is U, closely followed by
AA^ and N s- In fact, the difference between the results obtained with
n and AN gives one pause to consider if the added complexity of deter-
mining n is worth the relatively small gain in correlation with the radio
data. The results obtained with Qe and go (the N gradient of the initial
layer as reported by the radiosonde) are disappointingly small and are of
no practical significance when compared with those of H, AN, and even
A.,.
COMPARISON OF RADIO-METEOROLOGICAL PARAMETERS
Table 6.6. Summary of correlations
265
Radio-meteorological variable
Cate-
Sample
gory*
size
n
n'
n"
ge
9o
N,
AN
WENR.TV
(179.75 Mc/s)
1
68
-0. 3794
-0. 3795
-0. 1859
0. 1889
-0. 1884
0. 1414
-0. 3465
2
21
-. 2064
-. 2683
.2003
.0035
.1346
.0531
-.2653
3
101
-.1748
-. 1754
.0087
.0291
-. 0303
-. 0940
-.4553
4
6
-.3632
-.3503
-.4315
.4990
-.4149
.0905
.0873
5
196
-. 5665
-. 5732
-.0312
.0555
-.1772
.3870
-. 5640
6
518
-.4624
-. 4602
-.0169
.0722
-. 1762
.2889
-. 4544
WBKB-TV
(71.75 Mc/s)
1
67
-.2952
-.2976
-.2231
-.0926
.0791
.0107
-. 1687
2
27
-. 2071
-. 1843
-. 1378
-. 1037
.0939
.1678
-. 1608
3
113
-. 2691
-.2709
.0763
.1652
-.0810
.0703
-. 3486
4
6
-. 2445
-. 2202
-.6849
.3084
-.3686
.2573
.3485
5
213
-.5519
-. 5562
-. 0576
.0624
-.1152
.4168
-. 4635
6
564
-. 4384
-.4398
.0087
.0430
-. 1027
.2128
-. 3505
♦Category
l = Unstratifled.
2= Ground-based duct.
3 = Elevated layer.
4 = Subrefractive layer.
5=1 through 4 combined.
6= All data for period of record.
The above data also allow one to evaluate a parameter of the form
H, = age-^ bW,
which is analogous to Misme's recently suggested M parameter [19]. The
parameter Hi given above is physically more realistic for radio purposes
since it incorporates a mea.sure of the refractive index turbulence rather
than thermal stability. The correlations obtained between field strength
and Hi are
WENR-TV:
WBKB-TV :
0.46
0.44
which represents, for the significant figures carried, no improvement over
the use of 11 alone. This last analysis suggests that it might be desirable
to consider other combinations such as
H. = a.AN + 62 n.
The correlations for this case are:
WENR-TV: 0.49
WBKB-TV: 0.44
which represents a slight improvement over IT alone.
266 TRANSHORIZON PARAMETERS
One might well wonder why the correlations of Qe and field strengths
yield a higher correlation relative to that with AA'^ or A^s on a monthly
median basis than on a daily basis. This is apparently due to the non-
linear nature of the averaging process of g e which is dependent upon the
nonlinear weighting of the N{h) profile inherent in the refraction process.
On the other hand, the average AA'^ or A^^^ obtained from the average N{h)
curve is the same as that obtained from averaging AA'' or N s. Correla-
tions on n and field strength on a monthly median basis are now under
study.
6.3.6. Conclusions
In conclusion, we reach the opinion that the results of previous experi-
ence with the dependence of radio refraction upon N s and the concept
of stability are incorporated into IT. The resultant correlation with
radio data on a 3-hourly median basis is perhaps the most encouraging
obtained to date, but still, appears to be a marginal improvement over
the time-honored, and much simpler, parameter AA'^. This conclusion
could well change with broader experience on other paths and radio
frequencies.
6.4. References
[1] Bean, B. R., and F. M. Meaney (Oct. 1955), Some applications of the monthly
median refractivity gradient in tropospheric propagation, Proc. IRE 43, No.
10, 1419-1431.
[2] Bean, B. R., and B. A. Cahoon (Jan.-Feb. 1961), Correlation of monthly median
transmission loss and refractive index profile characteristics, J. Res. NBS 65D
(Radio Prop.), No. 1, 67-74.
[3] Misme, P. (Mar .-Apr. 1960), Le gradient equivalent mesure direct et calcul
theorique, Ann. Telecommun. 15, No. 3-4, 93.
[4] Misme, P. (Nov.-Dec. 1960), Quelques aspects de la radio-climatologie, Ann.
Telecommun. 15, No. 11-12, 266.
[5] Misme, P. (Jan.-Feb. 1961), Essais de radio climatologie dans le bassin du
Congo, Ann. Telecommun. 16, No. 1-2, 29.
[6] Flavell, R. G., and J. A. Lane (Jan. 1963), The application of potential refractive
index in tropospheric wave propagation, J. Atmos. Terrest. Phys. 24, 47-50.
[7] Moler, W. F., and W. A. Arvola (Aug. 1956), Vertical motion in the atmosphere
and its effect on VHF radio signal strength, Trans. Am. Geophys. Union 37,
399-409.
[8] Moler, W. F., and D. B. Holden (Jan.-Feb. 1960), Tropospheric scatter propaga-
tion and atmospheric circulations, J. Res. NBS 64D (Radio Prop.), No. 1, 81-93.
[9] Rice, P. L., A. Longley, and K. A. Norton (1959), Prediction of the cumulative
distribution with time of ground wave and tropospheric wave transmission
loss. Part I, NBS Tech. Note 15.
[10] Pickard, G. W., and H. T. Stetson (Jan. 1950), Comparison of tropospheric
reception, J. Atmos. Terrest. Phys. 1, 32-36.
[11] Bonavoglia, L. (Dec. 1958), Correlazione fra fenomeni meteorologici e propa-
gazione altre I'orizzonte sul Mediterraneo, Alta Frequenza 27, 815.
REFERENCES 267
[12] Gray, R. E. (Sept. 1961), Tropospheric scatter propagation and meteorological
conditions in the Caribbean, IRE Trans. Ant. Prop. AP-9, 492-496.
[13] Onoe, M., M. Hirai, and S. Niwa (Apr. 1958), Results of experiment of long-
distance overland propagation of ultra-short waves, J. Radio Res. Labs. 5, 79.
[14] Bean, B. R., L. Fehlhaber, and J. Grosskopf (Jan. 1962), Die Radiometeorologie
und ihre Bedeutung fiir die Ausbreitung der m-, dm-, and cm-Wellen auf
grossen Enternungen, Nachrichtentechnische Zeit. 15, 9-16.
[15] Bean, B. R., and G. D. Thayer (May 1959), On models of the atmospheric
refractive index, Proc. IRE 47, No. 5, 740-755.
[16] Misme, P. (Nov.-Dec. 1958), Essai de radiocHmatologie d'altitude dans le nord
de la France, Ann. Telecommun. 13, No. 11-12, 303-310.
[17] Bean, B. R., (Mar. 1962), The radio refractive index of air, Proc. IRE 50, No. 3,
260-273.
[18] Lane, J. A. (Feb.-Mar. 1961), The radio refractive index gradient over the
British Isles, J. Atmos. Terrest. Phys. 21, No. 2-3, 157-166.
[19] Misme, P. (May-June 1961), L'influence du gradient equivalent et de la stabilite
atmospherique dans les liaisons transhorizon au Sahara et au Congo, Ann.
Telecommun. 16, No. 5-6, 110.
[20] Boithias, L., and P. Misme (May-June 1962), Le gradient equivalent; nouvelle
determination et calculgraphique, Ann. Telecommun. 17, No. 5-6, 133.
[21] Saxton, J. A. (Sept. 1951), Propagation of metre radio waves beyond the normal
horizon, Proc. lEE 98, 360-369.
[22] Friis, H. T., A. B. Crawford, and D. C. Hogg (May 1957), A reflection theory for
propagation beyond the horizon, Bell Syst. Tech. J. 36, 627.
[23] duCastel, F., P. Misme, A. Spizzichino, and J. Voge (1958-1960), Resultats
experimentaux en propagation tropospherique transhorizon (a series of 10
papers), Ann. Telecommun. 13, 14, and 15.
[24] Craig, R. A., I. Katz, R. B. Montgomery, and P. J. Rubenstein (1951), Meteor-
ology of the refraction problem. Book, Propagation of Short Radio Waves by
D. E. Kerr, pp. 198-199 (McGraW-Hill Book Co., Inc., New York, N.Y.).
[25] Jehn, K. H. (June 1960), The use of potential refractive index in synoptic scale
radio meteorology, J. Meteorol. 17, 264.
[26] Bean, B. R., L. P. Riggs, and J. D. Horn (Sept.-Oct. 1959), Synoptic study of the
vertical distribution of the radio refractive index, J. Res. NBS 63D (Radio
Prop.), No. 2, 249-258.
[27] Misme, P. (1962), Private communication.
[28] duCastel, F., P. Misme, and J. Voge (1960), Sur le role des phenomenes de
reflexion dans la propagation louitaine des ondes ultracourtes, Electromagnetic
Wave Propagation, Book, p. 671 (Academic Press, London and New York).
[29] Booker, H. G., and W. E. Gordon (Apr. 1950), A theory of radio scattering in the
troposphere, Proc. IRE 38, No. 4, 401-412.
[30] Villars, F., and V. F. Weisskopf (Oct. 1955), On the scattering of radio waves by
turbulent fluctuations in the atmosphere, Proc. IRE 43, No. 10, 1232-1239.
[31] Norton, K. A., P. L. Rice, H. B. Janes, and A. P. Barsis (Oct. 1955), The rate of
fading in propagation through a turbulent atmosphere, Proc. IRE 43, No. 10,
1341-1353.
[32] Smyth, J. B., and L. G. Trolese (Nov. 1947), Propagation of radio waves in the
lower atmosphere, Proc. IRE 35, No. 11, 1198-1202.
[33] Booker, H. G., and W. Walkinshaw (1947), The mode theory of tropospheric
refraction and its relation to wave guides and diffraction. Book, Meteorological
Factors in Radio Wave Propagation, pp. 80-127 (The Physical Society,
London, England).
268 TRANSHORIZON PARAMETERS
[34] Gossard, E. E., and L. J. Anderson (Apr. 1956), The effect of super refractive
layer on 50-5000 Mc nonoptical fields, IRE Trans. Ant. Prop. AP-4, No. 2,
175-178.
[35] Starkey, B. J., W. R. Turner, S. R. Badcoe, and G. F. Kitchen (Jan. 1958), The
effects of atmospheric discontinuity layers up to the tropopause height on
beyond-the-horizon propagation phenomena, Proc. lEE, Pt. B, 105, Suppl.
8, 97.
[36] Abild, V. B., H. Wensien, E. Arnold, and W. Schilkorski (1952), tJber die Aus-
breitung ultrakurzer Wellen jenseits des Horizontes unter besonderer Beruck-
sichtigung der meteorologischen Einwiskungen, Tech. Hausmitteilungen des
Nordwestdeutschen Rundfunks, p. 85.
[37] Northover, F. H. (Feb. 1952), The anomalous propagation of radio waves in the
1-10 metre band, J. Atmospheric Terrest. Phys. 2, 106-129.
[38] Lane, J. A., and R. W. Meadows (Jan. 1963), Simultaneous radar and refrac-
tometer soundings of the troposphere. Nature 197, 35.
[39] Brekhovskikh, L. M. (1960), Waves in layered media. Book (Academic Press,
New York and London, England).
[40] duCastel, F. (1961), Propagation tropospherique et faisceaux hertziens trans-
horizon. Book, p. 90 (Editions Chiron, Paris).
[41] Wagner, N. K. (1960), An analysis of radiosonde effects on measured frequency
of occurrence of ducting layers, J. Geophys. Res. 65, 2077-2085.
[42] Bauer, J. R., and J. H. Meyer (Aug. 1958), Microvariations of water vapor in the
lower troposphere with applications to long-range radio communications,
Trans. Am. Geophys. Union 39, 624.
[43] Saxton, J. A. (July- Aug. 1961), Quelques reflexions sur la propagation des ondes
radioelectrique a travers la troposphere, L'Onde Elect. 40, 505.
[44] Hewson, E. W., and R. W. Longley (1944), Meteorology, Theoretical and Ap-
plied, Book, p. 352 (John Wiley & Sons, New York).
Chapter 7. Attenuation of Radio Waves
7.1. Introduction
The advent of tropospheric forward scatter techniques has made possi-
ble communications over longer distances with higher frequencies than
has been heretofore thought practicable. The limitations imposed by
gaseous absorption, and by scattering by raindrops, upon the power re-
quirements of a conmiunications system for this application become more
important with increasing distance and frequency. It has been common
in the past to evaluate propagation path attenuation due to absorption by
multiplying the ground separation of the terminals by the value of the
absorption calculated for surface meteorological conditions [1]^ or avoid
the problem by restricting the communications system to frequencies
that are essentially free of absorption [2]. This is in contrast to another
approach [3] which actually used the absorption along the ray path.
The following sections of this chapter will be devoted to a descriptive
treatment of absorption of radio waves by raindrops and gaseous oxygen
and water vapor in the atmosphere.
Unless otherwise specified, the following conditions will be assumed in
this chapter: (1) All attenuations will be expressed in terms of decibel
loss per unit length of the propagation path (dB/km). The attenuations
due to different causes are simply added to give the total attenuation in
decibels. (2) In this treatment average conditions of temperature, drop-
let size, and droplet distribution are assumed for the radio path in order
to approximate conditions met in practice.
7.2. Background
The attenuation experienced by radio waves is the result of two effects:
(1) absorption and (2) scattering. At wavelengths greater than a few
centimeters, absorption by atmospheric gases is generally thought to be
negligibly small except where very long distances are concerned. How-
ever, cloud and rain attenuation have to be considered at wavelengths
less than 10 cm, and are particularly pronounced in the vicinity of 1 and
3 cm.
' Figures in brackets indicate the literature references on p. 308.
2G9
270 ATTENUATION OF RADIO WAVES
It is helpful to recall that when an incident electromagnetic wave passes
over an object whose dielectric properties differ from those of the sur-
rounding medium, some of the energy from the wave is (a) absorbed by
the object and heats the absorbing material (this is called true absorp-
tion), and (b) some of the energy is scattered, the scattering being gener-
ally smaller and more isotropic in direction the smaller the scatterer is
with respect to the wavelength of the incident energy.
In the case of point-to-point radio communications we are interested
in the total attenuation of the scattering energy caused by losses resulting
from both the true absorption and the scattering.
7.3. Attenuation by Atmospheric Gases
The major atmospheric gases that need to be considered as absorbers
in the frequency range of 100 to 50,000 Mc/s are water vapor and oxygen.
For these frequencies the gaseous absorption arises principally in the
1.35 cm line (22,235 Mc/s) of water vapor and the series of lines centered
around 0.5 cm (60,000 Mc/s) of oxygen [4]. The variations of these
absorptions with pressure, frequency, temperature, and humidity are
described by the Van Vleck [4, 5] theory of absorption. The frequency
dependence of these absorptions is shown in figure 7.1 [4].
In connection with figure 7.1, the water vapor absorption values have
been adjusted to correspond to the mean absolute humidity, p, (grams of
water vapor per cubic meter) for Washington, D.C., 7.75 g/m^ The
reason for this adjustment is that water vapor absorption is directly pro-
portional to the absolute humidity [6] and thus variations in signal in-
tensity due to water vapor absorption may be specified directly in terms
of the variations in the absolute humidity of the atmosphere.
It can be seen from figure 7.1 that the water vapor absorption exceeds
the oxygen absorption in the frequency range 13,000 to 32,000 Mc/s,
indicating that in this frequency range, the total absorption will be the
most sensitive to changes in the water vapor content of the air, while
outside this frequency range the absorption will be more sensitive to
changes in oxygen density. Only around the resonant frequency corres-
ponding to X = 1.35 cm is the water vapor absorption greater than the
oxygen absorption. The absorption equations and the conditions under
which they are applicable have been discussed by Van Vleck [4], and the
best values to use for this section of the report have been taken from
Bean and Abbott [3].
The Van Vleck theory describes these absorptions from 100 Mc/s to
50,000 Mc/s in the following manner. The oxygen absorption at
ATTENUATION BY ATMOSPHERIC GASES
271
100 200 500 1,000 2,000 5,000 10,000 20,000 50,000
Frequency in Mc/s
Figure 7.1. Atmospheric absorption by the 1.35 cm line of water vapor and the 0.5 cm
line of oxygen.
272
ATTENUATION OF RADIO WAVES
T = 293 °K and standard atmospheric pressure in decibels per kilometer,
7i, is given by the expression:
Ti
0.34
Ai'i 1 Aj^2
X2
+
ji + AA (2 + iy + A.l
1
Al^2
(-
- 0' + ^'i
(7.1)
where X is the wavelength for which the absorption is to be determined
and where Ai'i and Ai'2 are line-width factors with dimensions of em~^
This formula is based on the approximations of collision broadening theory.
This theory postulates that, although the electromagnetic energy is
freely exchanged between the incident field and the molecules, some of
the electromagnetic energy is converted into thermal energy during
molecular collisions and thus a part of the incident electromagnetic energy
is absorbed. The term in (7.1 )involving Aj'i gives the nonresonant
absorption arising from the zero freciuency line of oxygen molecules while
the terms involving Az^2 describe the effects of the several natural resonant
absorptions of the oxygen molecule which are in the vicinity of 0.5 cm
wavelength. The (2 ± 1/X) (cm~^) terms are the portion of the shape
factors that describe the decay of the absorption at frequencies away from
the resonant frequency (the number 2 is the reciprocal of the centroid
resonant wavelength 0.5 cm).
The water vapor absorption at 293 °K arising from the 1.35 cm line,
72, is given by:
72
P
3.5 X 10"
X2
AV3
AX 1.35/
+ AV3
+
Avs
\X ^ 1.35/
+ AV3
(7.2)
where p is the absolute humidily and Afg is the line width factor of the
1.35 cm water vapor absorption line. The additional absorption arising
from absorption bands above the 1.35 cm line, 73, is described by:
73
P
0.05 Av4
X2
(7.3)
ATTENUATION BY ATMOSPHERIC GASES
273
where Au^ is the effective Hue width of the absorption bands above the
1.35-c'm Hne. The nonresonant term has been increased by a factor of
4 over the original Van Vleck formula in order to better satisfy experi-
mental results [7].
Although Van Vleck gives estimates of the various line widths, more
recent experimental determinations were used whenever possible. The
line-width values used in this chapter are summarized in table 7.1.
Table 7.1. Line width factors used to determine atmospheric absorption
Line
width
Temperature
Value
Sources
Al'2
Ai's
Aj/4
293 °K
300 °K
318 °K
318 °K
0. 018 cm-i atm^i
. 049 cm~i atm-i
. 087 cm^i atm-i
. 087 cm-i atm-i
Birnbaum & Maryott [8]
Artman & Gordon [9]
Becker & Autler [7]
Becker & Autler [7]
The preceding expressions for gaseous absorption are given as they
appear in the literature and do not reflect the pressure and temperature
sensitivity of either the numerical intensity factor or the line widths.
This sensitivity must be considered for the present application since it is
necessary to consider the manner in which the absorption varies with
temperature and pressure variations throughout the atmosphere. The
dependence of intensity factors upon atmospheric pressure and tempera-
ture variations was considered to be that given by the Van Vleck theory.
The magnitude and temperature dependence of the line widths is a
question not completely resolved. Both theory and experiment indicate
the line width to vary as 1/7'^, x <0. Different measurements on the
same line of oxygen have given values of x ranging from 0.71 to 0.90
with differences in the magnitude of Ai^ of about 2 percent [10, 11]. Ex-
periments have also clearly indicated that the line width changes from
line to line, with maximum fluctuations of about 15 percent. In the
frequency region considered in this chapter (10 to 45 Gc/s) the centroid
frequency approximation for oxygen is valid and a mean line width can be
used with good accuracy, but in the region of the resonant frequencies of
oxygen, the line-to-line line width variations must be taken into account.
The expressions used to calculate the absorptions are given in table 7.2.
The reference temperatures given are those at which the appropriate
experimental determinations were made, and the pressures are to be
expressed in millibars. A detailed discussion of the theoretical aspects of
the pressure and temperature dependence is given by Artman [12].
Experimental measurements on the absorption of microwaves by the
atmosphere (performed after our original work) show different values of
the loss than those obtained by theoretical prediction methods. There
is reasonably good agreement between the predicted and measured loss
for oxygen, but the measured loss of water vapor is considerably greater
274 ATTENUATION OF RADIO WAVES
Table 7.2. Values used in the calculation of atmospheric absorption
Absorption
Intensity factor Line width
dB/km
71
0. 34 / P \ /293\ 2
X2 \1013. 25/ \ T J
cm '
\1013. 25/ \ T /
and
\1013.25/ \ T /
72*
P
0. 0318 /293\ 5/2 / 644\
X2 \ T / \ T /
/ P \ /318\ "2
Ai-a ( ) { ) (1+. 0046p)
\1013. 25/ \ T /
73*
p
0. 05 /293\
X2 \ T/
/ P \ /318\>/2
Ar4 ( ) ( — ) (1+. 0046p)
\1013. 25/ \ T /
*p is water vapor density in g/m^
than that of the predicted amount, particularly above 50,000 Mc/s [13].
These observed discrepancies have little effect upon the present study,
which is confined to frequencies less than 50 Gc/s. The results of the
present study, for the frequency range 100 to 50,000 Mc/s, agree with
those reached by Tolbert and Straiton [14] in their field experiments at
Cheyenne Mountain and Pikes Peak, Colo., at altitudes of 14,000 ft.
The above approach represents that presented by Bean and Abbott
[3]. The following treatment was given by Gunn and East [15] and
based on Van Vleck's two papers [4, 5]. This latter presentation is only
valid when single line absorption with no appreciable overlap from
adjacent lines is considered.
By taking into account the temperature and pressure dependence of
the line widths, it is seen that for a given quantity of water vapor, the
attenuation is proportional to
P- and T- exp (- f ) ,
at the resonance line, to
P and T~' exp (- ^)
at the sides of the curve, and to P and T"3/2 ^g][ away from resonance.
In applying the above considerations to absorption approximations it
also must be remembered that for a given relative humidity, the density
will vary considerably with temperature. Table 7.3 shows attenuation
by water vapor at various temperatures and wavelengths.
ATTENUATION BY ATMOSPHERIC GASES 275
The behavior of water vapor attenuation near the resonant line is very
remarkable, as can be seen by inspecting (7.2). Since Av^ is small com-
pared to 1/X, it may be neglected in the denominator of (7.2) for non-
resonant wavelengths. The attenuation per unit density is thus directly
proportional to /^v^. and hence to the total pressure for these frequencies.
But at the resonant frequency, the dominant term in the expression is
proportional to l/A^s, and thus inversely proportional to the pressure.
In the atmosphere, the water vapor density is proportional to the total
pressure. Therefore, the attenuation is independent of pressure at the
resonant frequency and now depends only on the fraction of water vapor
present. For practical purposes, this means that attenuation can occur
at high altitudes with the same effectiveness as in the lower, denser layers
if the mixing ratio is the same.
On the other hand, oxygen absorption occurs because of a large number
of lines around 60 Gc/s. In the region from 3 to 45 Gc/s the attenuation
is proportional to P- and to T-n/^ ^5j ^g ^j^g temperature decreases the
attenuation increases gradually. At —40 °C oxygen attenuation is about
78 percent higher than at 20 °C due to increased density at low tempera-
tures. Table 7.4 shows the pressure and temperature corrections for
oxygen attenuation at wavelengths between 0.7 and 10 cm.
Figure 7.2 shows the attenuation at a pressure of 1 atmosphere and
20 °C as a function of wavelength [15]. The solid lines represent values
of attenuation measured by Becker and Autler [7]. The dashed line
shows values calculated from Van Vleck's theory. The water vapor
absorption curve, c, corresponds to a water content of 1 g/m^.
Since absorption is so sensitive to the absolute humidity level, it is
helpful to have information on the climatic variation of absolute humidity
throughout the 1 to 99 percent range of values normally used in radio
engineering. Estimates of the values of absolute humidity at the surface
expected 50 percent of the time for the United States for February and
August are given in figures 7.3 and 7.4 respectively [16]. It is evident
that for either month the coastal regions display greater values of absolute
humidity than do the inland regions. Note that for any location the
August values are consistently greater than the February values. Figures
7.5 through 7.8 show the values of absolute humidity expected to be
exceeded 1 and 99 percent of the time throughout the United States in
both summer and winter.
In addition to oxygen and water vapor, there are a number of other
atmospheric gases which have absorption lines in the microwave region
from 10 to 50 Gc/s. These gases normally constitute a negligible portion
of the general composition of the atmosphere, but could conceivably
contribute to attenuation. Table 7.12 shows the resonant freciuencies,
maximum absorption coefficients at 300 °K (attenuation coefficient if the
fraction of molecules present were equal to unity), expected concentration
276
ATTENUATION OF RADIO WAVES
Table 7.3. Water vapor attenuation (one way) in dB/km
[After Gumi-Eastl.
P, pressure in atmospheres; W, water vapor content in g/m"^
T
(°C)
X(cm) 10
5.7
3.2
1.8
1.24
0.9
20
-20
-40
0.07X10-3PW
0. osxio-^pw
0.09X10-3PW
0. loxirspw
0. 24X10-3P\V
0.27X10-3PW
0.30X10-3PW
0.34X10-3PW
0.7X10~3PW
O.SXIO-'PW
0.9X10-3PW
1.0X10-3PW
4.3X10-3PW*
4. 8X10-3PW*
5. OXlO-spW
5. 4X10-3PW
22.0X10-3PW*
23.3X10-3PW*
24.6X10-3PW'
26. 1X10-3PW*
9. 5X10-3PW
10.4X10-3PW
11. 4X10-3PW
12. 6X10-3PW
*The pressure dependencies shown are only approximate. Near the 1. 35 cm water vapor absorption line
(between 1. cm and 2. cm) no simple power dependency of P and W is accurate.
Table 7.4. Pressure and temperature correction for oxygen attenuation for wavelengths
between 0.7 and 10 cm
[After Qunn-East]
n°c)
Factor
(P is pressure in atmospheres)
20
1.00P2
1. 19 P2
-20
1. 45 P2
-40
1. 78 P2
o
O I0-' r
CVJ
<
I 10-2
CD
8 10-3
<
Z
UJ
^-4
fV
\
A ">'
7
\i , .
:^>s^
"^^^(<')
-
\
S
.
\
\
-
X (C)
*
\
•
s
v
\
\
■
\
V
1 1
1 1 1 . i .. I 1 'Ik 1
0.8
.5 2
6 8 10
A (cm)
Figure 7.2. Attenuation of microwaves by atmospheric gases.
(After Guim and East, 1954).
ATTENUATION BY ATMOSPHERIC GASES
277
Figure 7.3. Estimate of the value of absolute humidity expected 60 percent of the time
for February.
Figure 7.4. Estimate of the value of absolute humidity expected 50 percent of the time
for August.
278
ATTENUATION OF RADIO WAVES
Figure 7.5. Values of absolute humidity expected to be exceeded 1 percent of the time
for February.
Figure 7.6. Values of absolute humidity expected to be exceeded 1 percent of the time
for August.
ATTENUATION BY ATMOSPHERIC GASES
279
Figure 7.7. Values of absolute humidity expected to be exceeded 99 percent of the time
for February.
Figure 7.8. Values of absolute humidity expected to be exceeded 99 percent of the time
for August.
280 ATTENUATION OF RADIO WAVES
in the atmosphere, and expected absorption coefficients due to these
trace constituents. The data on molecular absorption coefficients was
taken from Ghosh and Edwards [33], that on concentrations from the
Glossary of Meteorology [29]. It is readily seen that the attenuation due
to these sources is negligible compared to the high absorption due to
oxygen and water vapor.
7.4. Estimates of the Range of Total
Gaseous Absorption
The range in gaseous absorption can be seen by considering the data
for the months of February and August at Bismarck, N. Dak., and
Washington, D.C., two stations with very different climates. The values
of total gaseous absorption (defined as the sum of 71, 72, and 73, where
7i = oxygen absorption in decibels per kilometer, 72 = water vapor
absorption arising from the 1.35 cm line and 73 = additional absorption
arising from absorption lines whose frequencies are considerably higher
than that corresponding to the 1.35 line) at each station and elevation up
to 75,000 ft are shown in figures 7.9 and 7.10 for each of the four station
months for the frequency range of 100 to 50,000 Mc/s. Above 75,000 ft
the absorption values for all four station months are identical and are
given for each frequency in figure 7.11. The absolute humidity was
calculated using the upper air monthly average values of temperature,
pressure, and humidity as reported by Ratner [17]. Readings for the
relative humidity are not generally given in this report for altitudes
greater than about 15 km due to the inability of the radiosonde to meas-
ure the small amount of humidity present at these altitudes. It is
believed that the climates represented by these station months encompass
the range of those of the majority of the continental United States radio
propagation paths.
An interesting property of the annual range of absorption as a function
of the frequency may be seen in figures 7.9 and 7.10. For the first 5,000
ft above the surface, it is noted that in the frequency range of 10 to 32.5
Gc/s, the summer values are greater than the winter values due to in-
creased humidity of the summer months. Outside of this frequency
range, however, the winter values of absorption are greater due to the
increased oxygen density.
In the frequency range 6 to 45 Gc/s, atmospheric absorption, 7;,, at
a frequency v, arises primarily from oxygen absorption, jdi,, and water
vapor absorption, juw', i.e.,
7^ = 7rf. + Jwv. (7.4)
RANGE OF TOTAL GASEOUS ABSORPTION
281
7
5
3
2
0.1
a?
0.5
a3
0.2
0.1
0.07
ao5
0.03
0.02
0.01
0.007
0005
0003
0.002
ODOI
: : : ..
BISMARK, N D.
'
\
-\
^
-50,000 Mc/s
\
HJ
-
ebruary
ugust
A
\ ^
%v
\ N
\
\ 1 1 ._! ._
\,
\
1 i
\.
vV ^
^x-U 22,000 Mc/s
^^
\r\\^ ^
'v
\
32,000 Mc/
\\
\
\
\
\
\:-
q3
\
\
\
\
\
\
\
E
\^r
\ X \
HO,OOOMc/s
\
x\
s ^
—
\
^■^
\
\
i^
^
N
*^
\\
a>
^""C"^
\
\
\s
\N
\\
Iv
\
^N
3,000 Mc/s^^
\
I %
\
o
0)
Q
§^
\
\
\
^-
^.
\
o
^
tg^
00 Mc
/s ^
--^ i ^c^
\
\
\\
\
S
'^--'-''^' "S."^
^,,
\ V
^
^'V
>^ ^^J^.
\
\
<
SA^
,^
^ N
N
\
C'^v
N ^
\ ^.
^.\^'^
\
\
\'
V
NV\x
\ ^
\ ^v
y\
is^
A
\
\v \
'
\^
\
\
-T_^
— ^ — ^ —
■~>=».
\
-100 w
c/s
^^
..,__2l"^->
^v
k
\
X-^XXX \,\ ! N
k^^^^^ >^^
OBOOS
0M03
QD002
onooi
X^Sy^
V
>!&5-,
s
X
•^
^
'x
^
s
^
^
5 10 15 20 25 30 35 40 45 50 55
Height Above the Surface in Thousands of Feet
65 70 75
Figure 7.9. Total gaseous atmospheric absorption from the surface to 75,000 feel:
Bismarck, N. Dak.
282
ATTENUATION OF RADIO WAVES
1
1
1 1 1 :
■
1
0.T
^\J 1 W
'1 *-'•
0.5
vx.
'50,000 Mc/s
^
0.2
^\
'\
is.
-
August
■*v
^^
\
s V
\ \
V
]
\v
\
V
1
s \
^^
IZ,Z0(.
Mc/s
^
N
\
\,
^</'
\^
neter
s
n\
^ /
' <
\
N
?>^
\
\
\
s.
32,50
DMc/s
K^
^J^
\ ""n
\'^^
V ^s
000 M
vV
\
S ^
^VS
\
\
I \'
r
s ^
^ \
s N
^\
\
N. ^
\, ^
oi
^•,.\
N>
\ >!
\
\
^
\^
\.
NN.
•"§ aoo3
.£ 0.002
o
Q.
_§ 0.001
"^
^
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\
V
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\
V
3,a
DO Mc/
S^
^
V,
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'^
^ '^S.
\\
^^^
:r7f^
-300 r
Ic/S
^
k
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s ^,
0.0007
"^
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^\Si\"~"^v
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N
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aoooo5
^
V
>
%s
0.00001
^
5 10 15 20 25 30 3a 40 45 50 55
Height Above the Surface in Thousands of Feet
65 ID 75
Figure 7.10. Total gaseous atmospheric absorption from the surface to 75,000 feet:
Washington, D.C.
TOTAL RADIO PATH ABSORPTION 283
Zhevankin and Troitskii [31] have indicated that 7^^ and 7^,^ can be repre-
sented as exponential functions of height, Z, above the earth's surface,
7d. = 7d.o exp [j^~), Iw. = Tu-.o exp \j[—)> (7-5)
where 7<f„o and 7^;,o are the values of jdv and jwy, respectively, at the
earth's surface, and H dv and //„,;, are called the "scale heights" of 7^^ and
7^,„. This model is known as the "bi-exponential" model of absorption,
and ydy and 7,^,^ are often called the "dry" and "wet" terms of y^. The
scale height for the dry term in the frequency range 6 to 45 Gc/s can be
written as [32]
Hd. (m km) ^ -—V (7.6)
where To is the surface temperature in °K, a is the temperature lapse rate
with height in °K/km, and b, c, are constants determined from thermo-
dynamic considerations. Because of the hump in the Hy,^ curves as op-
posed to the flat H dp curve in figure 7.28, such a handy expression as (7.6)
for H „,^ is not possible in the 6 to 45 Gc/s frequency range (fig. 7.28 was
determined from actual radiosonde data at Verkhoyansk, U.S.S.R.).
7.5. Total Radio Path Absorption
The total path absorption is determined by calculating the various
absorption coefficients as functions of the heights along the ray path and
then numerically integrating the values along the entire path using stand-
ard ray tracing techniques outlined in chapter 3. The values of total path
integration over a 100-km path thus obtained are presented in figure 7.12
for Bismarck, N. Dak., and Washington, D.C. The difference between
the two climates is evident principally at the higher frequencies, where
the Washington absorptions are consistently above the Bismarck values.
This is apparently due to a combination of generally greater humidities
and greater refractive effects. These two effects are related. The in-
creased humidity at Washington enhances the water vapor absorption
and increases the refraction causing the radio ray to travel consistently
through lower levels of the atmosphere with consequent increase in total
path absorption.
284
ATTENUATION OF RADIO WAVES
aoooooo3
00000002
\
\
S
\
\
\
\
s
\
^50,0(
X) Mc/
s
\
\
V
k
\
\
N
\
\
\,
\
\
\
\
v^^
2,500 Mc/s
\
\
^
\
^
\
\
\
^^
^22,2
oo\
:/s
\
\
\
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V
\
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^N
\
N
N
^>
\
\
\
^
k
\
\
\
lopc
)OMc/s
\n
\
\
\
3,000
Mc/s '
^
\
300 £
ilOOM
c/.'^
k
\
\
\
\
\
V\,
\
\
X^
\
\
X
\
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\,
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\
s
Heigiit Above the Surface in Thousands of Feet
Figure 7.11. Common values of total gaseous atmospheric absorption for elevations
greater than 75,000 feet.
TOTAL RADIO PATH ABSORPTION
285
CD 20
CD I
500 700 1000
2000 5000 7000 10000
FREQUENCY (MC/S)
20000
50000 70000 100000
Figure 7.12. Total path absorption over a 1000 km propagation path with the climate
of Bismarck, N. Dak.
286 ATTENUATION OF RADIO WAVES
7.6. Derivation of Absorption Estimate for Other Areas
The values of total path absorption given above are for two specific
locations. For this particular study to be of practical use, a means
should be provided for arriving at estimates of geographic and annual
variations of total path absorption for various surface distances and
frequencies. The method chosen utilizes the correlation between total
path absorption and the surface value of the absolute humidity, expressed
in grams of water vapor per cubic meter, which appears explicity in
Van Vleck's water vapor absorption formulas [4,5]. The basis for expect-
ing a correlation to exist between the absolute humidity and the total
path absorption is that the absorption at those frequencies for which
water vapor absorption is dominant (approximately 10 to 32 Gc/s) varies
directly as the absolute humidity while, for those frequencies at which
oxygen absorption is dominant, it varies inversely as the absolute humid-
ity due to the inverse relationship of oxygen density and water vapor
density. That is, during the warm seasons of the year the total atmos-
pheric pressure tends towards its yearly minimum (as does the oxygen
absorption), while the absolute humidity tends towards its yearly maxi-
mum (as does the water vapor absorption). Conversely, during the
colder seasons of the year, the pressure tends towards its maximum value
while the absolute humidity tends towards its minimum value [3].
As an example of the correlation method, the surface absorption was
calculated at a water vapor-dominated frequency (22.2 Gc/s) and
oxygen-dominated frequency (50 Gc/s) for each month throughout the
year for both Washington and Bismarck. These values are plotted on
figure 7.13. The term surface absorption is used for the values of ab-
sorption calculated from standard ground level weather observations.
The water-dominated 22.2 Gc/s data fall on a smooth curve despite the
pressure and temperature differences of the two stations. The oxygen-
dominated 50 Gc/s data, however, display an interesting separation of
points for each station, although the distribution of points at the two
locations display similar slopes. The 50 Gc/s absorption is more sensi-
tive to the atmospheric density difference between the stations. If the
pressure differences were taken into account, the Bismarck data would
increase about 12 percent and the two curves would be distributed along
a common line w^ith the same slope as the original two curves. This
figure, then, indicates that the absorption is correlated with the absolute
humidity. The above illustration is for surface values rather than for
integrated propagation path values. Variations in the upper air meteor-
ology that are not reflected in the surface values will tend to diminish
the correlation. (For comparison of percentage absorption over a 300-km
path, the first few hundred feet contribute about 5 percent at 100 Mc/s
increasing to 42 percent at 10 Gc/s and remaining constant to 50 Gc/s.)
Keeping these reservations in mind, one may utilize the method of least
ABSORPTION ESTIMATE FOR OTHER AREAS
287
o
Q.
O
CO
0.07
0.05
0.03
0.02
0.01
0.007
0.005
0.003
0.002
0.001
"—
-^
&
--^
-^
>
>o
i»
/
/
/
/
/
%
/
/
/
r/
^
/
/
/
</
o Washington 0^ at 50,000 Mc/s
• Washington H^O at 22,200 i^c/s
9 Bismar/< 0^ at 50,000 l\4c/s
« BismariK H^O at 22,200 /l/c/s
1 1 1 1 1 1
0.2 0.3 0.4 0.5 OB 0.7 0.8 0.9
1.2 1.3
Log of Absolute Humidity (gm/m )
Figure 7.13. Surface value of 22.2 and 60 Gc/s absorption versus absolute humidity at
Washington, D.C., and Bismarck, N.Dak.
288
ATTENUATION OF RADIO WAVES
Table 7.5. Values of m and b in the regression line y = mx + b,
where y is the logarithm of the total path attenuation, y, b is the value of y when the absolute humidity
is 1 g/m^ and x is the logarithm of the absolute humidity.
[After Bean and Abbott]
Distance
100 km
300 km
1000 km
Freq. (Gc/s)
m
b
m
6
m
6
100 ..
-0. 07263
-. 08212
-. 10203
+. 06996
+. 26022
+. 48097
+. 62034
+. 75045
+. 77630
+.77153
+. 74979
+. 54155
+. 46305
+. 25936
+. 18460
-3. 69822
-1.72139
-0. 49427
-. 24077
-. 14038
+. 20891
+. 65324
+1. 41086
+1. 67001
+1. 69105
+1. 62838
+1.34408
+1. 51053
+2. 46361
+2. 91681
-0. 06324
-. 06363
-. 07078
+. 06872
+. 23447
+. 44214
+. 58044
+. 70693
+. 72874
+. 72508
+. 70659
+. 50126
+. 42416
+. 23331
+. 16668
-2. 60042
-0. 73037
+.31846
+. 54167
+. 64637
+. 98393
+1. 42477
+2. 25807
+2. 57270
+2. 57734
+2. 46633
+2.11161
+2. 28497
+3. 25085
+3. 70472
-0. 13298
-. 15276
-. 16932
-.04731
+. 10831
+. 31506
+.46112
+. 60558
+. 62885
+. 62963
+. 60458
+. 37634
+. 29646
+. 10712
+. 04360
1 83711
300
17792
1000
+ .75687
+ .96517
+ 1.06172
+1. 36994
+1. 77604
+2. 56360
+2. 87766
+2. 87575
+2. 77175
+2. 48455
+2. 67460
+3. 66659
+4. 12531
6000
10,000
15,000
18,000
21,000
22,406
23,076 .
24,000
30,000
33,000
42,000_ .
45,000
SWOO 70CWO ICBOM
Figure 7.14. Values of h and m in the regression line y = mx + b, where y is the
logarithyn of the total path absorption, x is the logarithm of the absolute humidity, b is
the value ofy when the absolute humidity is 1 g/m^.
ABSORPTION ESTIMATE FOR OTHER AREAS
289
squares to obtain the total path absorption as a linear function of the
absolute humidity. The parameter is a statistical regression line at each
frequency for path propagation distances of 100, 300, and 1,000 km as
given in table 7.5, and also plotted on figure 7.14, allowing the reader to
estimate the regression line for frequencies other than those of the
present study [3].
Statistics for the variation of absolute humidity have already been
given, and contours of the mean values of absolute humidity for the world
are given on figures 7.15 and 7.16 for the months of February and August
respectively. It is noted that the absolute humidity for either month is
greater in the coastal regions than in the continental interiors and that
the windward sides of continents have larger values than do the leeward
sides.
By reference to figures 7.15 and 7.16 [16] for the United States, and
figure 7.14, estimates of radio power loss due to absorption may be
obtained. As an example, figure 7.17 gives the values expected to be
exceeded 1 percent of the time over a 300 km propagation path at 10,000
Mc/s for the United States during the month of August.
»• lOO- nr \vr i60't»si iwiEsi lec no* i!0' lOO* sc
«• 20ESI 0'£«SI W «!■
Figure 7.15. Average absolute humidity (g/ni^), February.
290
ATTENUATION OF RADIO WAVES
20* 140' 160'iASI I80'*ESI leC HO* 120' W eo* 60' «• WKSI 0*E«SI JIf w
100' OT IW lSO'f«ST IJO'IESI ISO- HO- l!0' 100' IC »• 40* !<ri£ST 0*E»ST 20' 40*
Figure 7.16. Average absolute humidity {g/m}), August.
Figure 7.17. August values of 300-mi propagation path absorption loss {in dB) to be
exceeded 1 percent of the time for a frequency of 10,000 Mc/s.
ATTENUATION IN CLOUDS
7.7. Attenuation in Clouds
291
Cloud droplets are regarded here as those water or ice particles having
radii smaller than lOO/i or 0.01 cm. For wavelengths of incident radiation
well in excess of 0.5 cm, the attenuation becomes independent of the
drop size distribution. The generally accepted equations for attenuation
by clouds usually show the moisture component of the equations in the
form of the liquid water content (g/m^). Observations indicate that the
liquid water concentration in clouds generally ranges from 1 to 2.5 g/m^
[18], although Weickmann and aufm Kampe [19] have reported isolated
instances of cumulus congestus clouds with a reading of 4.0 g/m^ in the
upper levels. In ice clouds, it will rarely exceed 0.5 and is often less than
0.1 g/m^. The attentuation of cloud drops may be written as:
K = KiM,
where K = attenuation in dB/km,
Ki = attenuation coefficient in dB/km/g/m', and
M = liquid-water content in g/m^.
Values of Ki by ice and water clouds are given for various wavelengths
and temperatures by Gunn and East in table 7.6.
Table 7.6. One-way attenuation coefficient, Ki, in clouds in dB/km/gm/m^
Temperature
Wavelength (cm)
(°C)
0.9
1.24
1.8
3.2
Water cloud
Ice cloud
20
10
-8
-10
-20
0.647
.681
.99
1.25
8. 74X10-3
2.93X10-3
2. XlO-3
0.311
.406
.532
.684
6.35X10-3
2.11X10-3
1.45X10-3
0.128
.179
.267
.34
(extrapolated)
4.36X10-3
1.46X10-3
1.0 xio-3
0.0483
.0630
.0858
.112
(extrapolated)
2.46X10-3
8. 19X10-3
5.63X10-"
Several important facts are demonstrated by table 7.6. The decrease
in attenuation with increasing wavelength is clearly shown. The values
change by about an order of magnitude for a change of X from 1 to 3 cm.
The data presented here also show that attenuation increases with de-
creasing temperature. Ice clouds give attenuations about two orders of
magnitude smaller than water clouds of the same water content. The
attenuation of microwaves by ice clouds can be neglected for all practical
purposes [20]. The comprehensive works of Gunn and East [15] and
Battan [20] on attenuation offer excellent sources of detailed information
on this subject.
292 ATTENUATION OF RADIO WAVES
7.8. Attenuation by Rain
Ryde and Ryde [21] calculated the effects of rain on microwave propa-
gation and showed that absorption and scattering effects of raindrops
become more pronounced at the higher microwave frequencies where the
wavelength and the raindrop diameters are more nearly comparable. In
the 10 cm band and at lower wavelengths the effects are appreciable, but
at wavelengths in excess of 10 cm the effects are greatly decreased. It is
also known that suspended water droplets and rain have an absorption
rate in excess of that of the combined oxygen and water vapor absorp-
tion [3].
In practice it has been convenient to express rain attenuation as a
function of the precipitation rate, R, which depends on both the liquid
water content and the fall velocity of the drops, the latter in turn depend-
ing on the size of the drops.
Ryde studied the attenuation of microwaves by rain and .showed that
this attenuation in dB/km can be approximated by:
Ku=K RirYdT {1.1)
where Kr = total attenuation in dB
K = con.stant
R{r) = rainfall rate
r = length of propagation path in km, and
a = constant.
Laws and Parsons [22] observed the distribution of drop sizes for
various rates of fall on a horizontal surface. The higher the rainfall rate,
the larger the droits, and also the greater the range in size of the drops.
However, in order to derive the size distributions occurring while the
drops are falling in the air, each rainfall rate must be divided by the
particular velocity of fall appropriate to the corresponding drop diameter.
Figure 7.18 shows the resulting distribution in air, expressed as the rela-
tive mass of. drops of given diameter, for three chosen representative
values of precipitation rate, p, namely, 2.5 mm/hr, 25 mm/hr, and 100
mm/hr [23]. The excess path loss per mile, according to Ryde, for the
three carrier frequency bands of 4, 6, and 11 Gc/s is shown on figure 7.19.
Figure 7.20 [24] is a scatter diagram showing transmission loss versus rain-
fall rate. For comparison, Ryde's equation is plotted on figure 7.20.
The greatest uncertainty in predictions of attenuation due to rainfall,
using theoretical formulae as a ba.sis for calculation, is the extremely
limited knowledge of drop size distribution in rains of varying rates of
fall observed under differing climatic and current weather conditions.
ATTENUATION BY RAIN
293
w
w
<;
'^
w
>
I— I
H
<:
w
Pi
O-G
Figure 7.18. Relative total mass of liquid water in the air contributed by rain drops of
diameter D for various precipitation rates.
(Derived from Laws and Parsons' distributions for a horizontal surface by dividing by the appropriate
terminal velocities.)
294
ATTENUATION OF RADIO WAVES
40
20
1
0.8
0.6
0.4
0.2
O.t
0.08
0.06
0.04
0.02
O.Ot
/
/
/
/
/
/
/
/
/
/
/
/
/
^
r
/
/
Lfv
f
/
(
<;\
r
/
^
<0
y
/
/
/
J
/
/
f
r
/
/
/
/
J
f
j\
/
/
/
f
/
/
>
/
'
/
/
/
/
/'
/
/
/
/
/
_
0.01 0.02 0.04 0.1 0.2 0.4 0.6 1 2
RAINFALL RATE IN INCHES PER HOUR
Figure 7.19. Rain attenuation versus rainfall rate.
(Theoretical, after Ryde and Ryde).
6 8 10
There is little evidence that a rain with a known rate of fall has a unique
drop-size distribution though studies on this problem seem to indicate
that a most probable drop size distribution can be attached to a rain of
given rate of fall [22]. Table 7.8 shows the percentage of total volume
of rainfall occupied by raindrops of different diameters (cm) and varying
rainfall rates (mm/hr). Tables 7.8 and 7.9 are offered as an aid to
estimating, through a qualitative approach, the attenuation of radio
waves by raindrops [25].
Table 7.8 gives the decibel attenuation per kilometer in rains of different
rates of fall and radio wavelengths between 0.3 and 10 cm. In table 7.9,
ATTENUATION BY RAIN
295
similar to table 7.8 an additional set of results is contained for rains of
measured drop size distribution.
Since the total attenuation cross section [26] depends on the temperature
(because of its effects on the dielectric properties of water), it is important
to evaluate the attenuation of rains whose drops are at different tempera-
tures from those in the preceding tables. Table 7.10 contains the neces-
sary data relative to the change of attenuation with temperature and is
to be used with table 7.8. For example, in table 7.8, with a p^fcipitation
rate of p = 0.25 mm/hr, temperature of 18 °C, X = 1.25 cm, the attenua-
tion is 0.0215 dB/km. Using the correction factors obtained from table
7.10, for the same general conditions of precipitation and wavelength,
for a temperature reading of °C, the attenuation reads 0.02043 dB/km;
for a value of 30 °C an attenuation of 0.019350 dB/km is noted; and for
a temperature of 40 °C the attenuation is 0.01742 dB/km.
6.0
5.0
a 3.5
y
• >
/
• y *
•
•
/
' • •
t '
<
y
•
^
V
;
•
:>#
V"
•
'
•
/^
^.^
K^
c
y
■
•
^
V
!>
y
/
•(
^
2.0 2.5 3.0 3.5 4.0 5.0 6.0 7.0 8.0 10
TRANSMISSION LOSS IN DECIBELS PER MILE
Figure 7.20. Transmission loss in decibels per mile versus rainfall in inches per hour.
(After Hathaway and Evans, 1959).
296
ATTENUATION OF RADIO WAVES
Table 7.7. Drop size distribution
( Burrows-A ttwood )
Drop diameter,
D, in
centimeters
0.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
Precipitation rate, p, in mm/hr
0.25
1.25 2.5
Percentage of a given volume containing
drops of diameter, D
28.0
10.9
7.3
2.6
1.7
1.2
1.0
50.1
37.1
27.8
11.5
7.6
5.4
4.6
18.2
31.3
32.8
24.5
18.4
12.5
8.8
3.0
13.5
19.0
25.4
23.9
19.9
13.9
0.7
4.9
7.9
17.3
19.9
20.9
17.1
1.5
3.3
10.1
12.8
15.6
18.4
0.6
1.1
4.3
8.2
10.9
15.0
.2
0.6
2.3
3.5
6.7
9.0
.2
1.2
2.1
3.3
5.8
0.6
1.1
1.8
3.0
.2
0.5
.2
1.1
0.5
.2
1.7
1.0
0.7
1.0
4.1
7.6
11.7
13.9
17.7
16.1
11.9
7.7
3.6
2.2
1.2
1.0
0.3
Table 7.8. Attenuation in decibels per kilometer for different rates of rain precipitation
Temperature 18 °C
(Burrows-Attwood)
Precipi-
Wav
elength, X in cm
tation
rate, p,
in mm/hr
X = 0.3
X=0. 4
X = 0. 5
X = 0. 6
X = 1.0
X = 1.25
X=3.0
X = 3.2
X = 10
0.25
0.305
0.230
0.160
0.106
0.037
0. 0215
0. 00224
0.0010
0.0000997
1.25
1.15
0.929
0.720
0.549
0.228
0.136
0. 0161
0.0117
0. 000416
2.5
1.98
1.66
1.34
1.08
0.492
0.298
0. 0388
0. 0317
0. 000785
12.5
6.72
6.04
5.36
4.72
2.73
1.77
0.285
0.238
0.00364
25.0
11.3
10.4
9.49
8.59
5.47
3.72
0.656
0.555
0. 00728
50.
19.2
17.9
16.6
15.3
10.7
7.67
1.46
1.26
0. 0149
100.
33.3
31.1
29.0
27.0
20.0
15.3
3.24
2.80
0.0311
150.
46.0
43.7
40.5
37.9
28.8
22.8
4.97
4.39
0.0481
Table 7.9. Attenuation in rains of knoivn drop size distribution and rate of fall
(decibels per kilometer)
(Burrows-Attwood)
Precipitation
Wavelength X in cm
rate, p, in
mm/hr
1.
25
3
5
8
10
15
2.46
1.93
10-1
4.92
10-2
4. 24 10-3
1.23
10-3
7.34
10-*
2.80
10-*
4.0
3.18
10-1
8.63
10-2
7.11 10-3
2.04
10-3
1.19
10-3
4.69
10-4
6.0
6.15
10->
1.92
10-1
1. 25 10-2
3.02
10-3
1.67
10-3
5.84
io-<
15.2
2.12
6.13
10-1
5. 91 10-2
1.17
10-2
5.68
10-3
1.69
10-3
18.7
2.37
8.01
10-1
5. 13 10-2
1.10
10-2
6.46
10-3
1.85
10-3
22.6
2.40
7.28
10-1
5. 29 10-2
1.21
10-2
6.96
10-3
2.27
10-''
34.3
4.51
1.28
1. 12 10 1
2.32
10-2
1.17
10-2
3.64
10-3
43.1
6.17
1.64
1. 65 10-1
3.33
10-2
1.62
10-2
4.96
10-3
RAINFALL ATTENUATION CLIMATOLOGY 297
Table 7.10 Correction factor {multiplicative) for rainfall attenuation
(Burrows-Attwood)
Precipitation
rate, p, in
X cm
0°C
10 °C
18 °C
30 °C
40 °C
mm/hr
0.25
0.5
0.85
0.95
1.0
1.02
0.99
1.25
.95
1.00
1.0
0.90
.81
3.2
1.21
1.10
1.0
.79
.55
10.0
2.01
1.40
1.0
.70
.59
2.5
0.5
0.87
0.95
1.0
1.03
1.01
1.25
.85
.99
1.0
0.92
0.80
3.2
.82
1.01
1.0
.82
.64
10.0
2.02
1.40
1.0
.70
.59
12.5
0.5
0.90
0.96
1.0
1.02
1.00
1.25
.83
.96
1.0
0.93
0.81
3.2
.64
.88
1.0
.90
.70
10.0
2.03
1.40
1.0
.70
.59
50.0
0.5
0.94
0.98
1.0
1.01
1.00
1.25
.84
.95
1.0
0.95
0.83
3.2
.62
.87
1.0
.99
.81
10.0
2.01
1.40
1.0
.70
.58
150.
0.5
0.96
0.98
1.0
1.01
1.00
1.25
.86
.96
1.0
0.97
0.87
3.2
.66
.88
1.0
1.03
.89
10.0
2.00
1.40
1.0
0.70
.58
7.9. Rainfall Attenuation Climatology
The above paragraphs have been concerned with a descriptive presen-
tation of the theoretical and technical background of the problem of
power loss due to attenuation by rain and atmospheric gases. In an
attempt to circumvent the difficulties of the above methods of attenua-
tion prediction it was considered important to try a climatological ap-
proach to this problem. However, the results of such a study were
disappointing due to the fact that the problems of a systematic climato-
logical estimation of rainfall attenuation are many and varied. Answers
are needed for such questions as; how often do various rainfall rates and
drop sizes occur in geographical areas and over how large an area do
these rate and drop size statistics apply? Furthermore, to what height
in the atmosphere do these data apply [1]?
Unfortunately, the present state of meteorological knowledge concern-
ing these problems in such that no conclusions can be drawn on a syste-
matic and climatological basis. In view of these facts, it appears
prudent at this time to provide only engineering estimates of the com-
bined gaseous and rain absorption.
In this regard, Bussey [1] has shown that the absorption due to rainfall
exceeds that of gaseous constituents about 5 percent of the time for
frequencies around 6,000 Mc/s. The 5 percent figure was obtained by a
study of the rainfall rate distribution for various locations in the United
States. Figure 7.21 shows the combined rain and gaseous absorption
to be exceeded 1 per cent of the time.
298
ATTENUATION OF RADIO WAVES
bl).Ol)l)
n
n
~
"T
—
-
^
p=n
a
^
TTTl
/
<
zaooo
/
y
'
II
^
=»»
«•
z.
U--
=^
^
10.000
\tOOmite\^
7.000
^ ' :
/^ ■ >>-
y
Vi'^
y
<ry
f 1
II
3,000
/
/I
-^303/7,,*!
/
//
2.000
1.000
1,000 mile]
/
/i
^
100
II
j
//
500
'
//
/
'/
500
200
100
/
/
/
y
/
/
Y
^
-^
./■
^
:^
^
OjOI OjO! 01)3 01)5 007 0.1 02 0.3 05 0.7 I 2 3 5 7 10 20 30 50 70 100 200 300 500 TO IMC
Total Path Attenuation in Decibels
Figure 7.21. Combined rain and gaseous absorption to be exceeded 1 percent of the time.
7.10. Rain Attenuation Effects on Radio
Systems Engineering
Attenuation due to rainfall is obviously a dominating factor in deter-
mining the reliability of a communications system, especially at frequen-
cies in excess of 30 Gc/s. Rain varies greatly in frequency of occurrence
from one region to another, so it is important to have an effective method
of predicting the performance of a radio system in any region in order
that the communications engineer will be able to gain the widest possible
application and degree of reliability, consistent with cost, in any system
of his design.
This section will be concerned, in the main, with the results of the
Bell System's [24] field experiment in the Mobile, Ala., area, which was
designed to establish a relationship between excess path attenuation and
instantaneous rate of rainfall and to seek out any relationship between
the profile of rate of rainfall along a radio path and rainfall measured
at a point.
The main problem concerns the ability to predict outage time due to
rainfall (time the system noise exceeds the system objective) at 11 Gc/s
in all areas of the country. This is obviously a difficult problem since,
due to reasons of cost, it is not feasible to measure rainfall attenuation
in all parts of the country. Therefore, it is desirable to be able to use
what rainfall data are available and to couple the data, through what are
thought to be reasonable assumptions, to the relationships between rain-
fall and attenuations.
RAINFALL ATTENUATION CLIMATOLOGY
299
In approaching the problem of predicting the outage time due to rain-
fall, it has been assumed that the annual distribution of one-hour point
rainfall rates is indicative of the instantaneous values over 30-mi radio
paths [1] and that the frequency of severe rainfall of the type measured
in the Mobile area will be reduced in other parts of the country in propor-
tion to the distribution of annual point rates of 1 in. or more rainfall
per hour.
Figure 7.22 indicates the expected outage time due to rainfall for
various path lengths in different rain areas of the United States. The
curves A through H correspond to the areas contoured in figure 7.23
which illustrates contours of constant path length for fixed outage times
for different areas of the United States. The longer paths have been
somewhat weighted to take into account the less severe rainfall covering
larger geographical areas than the intense storms typical of the Gulf
region.
In a complete 11-Gc/s Bell System point-to-point relay system, the
rain outages of the individual hops must be added to obtain the perform-
ance of the system. It is desirable that the individual hops meet the
same objective, but this is not always possible. Sometimes one or more
//
1
/
/
/
/
/ /
/
/
/
/
A
/
a/
/ ^
/,
/
'./
7
/
/
/
/
/
1
/
/
/
A
//
/
y^
/
/
/
^/
/
^
^
c
^^ <«
^A
^
/
15 20 26
PATH LENGTH IN MILES
35 40 45 50
Figure 7.22. Expected outage time in hours per year versus path length in miles for
various areas of the United States.
(After Hathaway and Evans, 1959).
The curves A through H correspond to the areas contoured in figure 7.23.
300
ATTENUATION OF RADIO WAVES
Figure 7.23. Contours of constant path length for fixed outage time.
(After Hathaway and Evans, 1959).
hops of a system are electrically long; they will have insufficient fading
margin (the number of dB the receiver input level can be reduced before
the noise exceeds the system objective) and hence contribute more than
their share of the outage time. So, this excess must be made up by
imposing tighter requirements on the remaining hops. To meet the
overall objective of the Bell System, it is necessary to know the contri-
butions of the long hops — those having a fading margin less than 40 dB.
Figure 7.24 shows the excess path loss due to rain, versus hours per year
for the Mobile area study. Since the shape of this curve is nearly
identical to Bussey's curve of cumulative distribution for point rates in
Washington, D.C. (if we assume the shape of this curve to be representa-
tive for other areas of the country), then the additional outage time for
path lengths given by figure 7.23 can be estimated for hops having a
fading margin less than 40 dB. The data shown on figure 7.24 have been
rationalized and are shown in figure 7.25 as an estimate of additional
outage time.
Sometimes it is practical to shorten a proposed path to bring the fading
margin up to 40 dB. An approximation of the necessary reduction path
length can be made if uniform rainfall rate is assumed over the path.
RAINFALL ATTENUATION CLIMATOLOGY
301
^43
40
35
If)
_i
LU
ID
U 30
LU
Q
Z
If) 25
If)
O
_l
I 20
t-
in
If) 15
LU
u
X
LU
10
5
^
y^
^\^^
\
V)
V
V,
N
\
V
^^
N
N
s
NJ
\
6 7 8 9 10 20 30 40 50 60 70 80 100
HOURS THAT THE ORDINATE VALUE WAS EXCEEDED
200
Figure 7.24. Excess path loss due to rainfall versus hours per year at Mobile, Ala.
(After Hathaway and Evans, 1959).
40
35
Z
o
cr
< 30
o
2 25
Q
<
^ 20
15
20
30
40 50 60 70 80 100
200
300 400
Figure 7.25. Additional outage time expected for 11 Gc/s systems having a fading
margin less than 40 dB.
302
ATTENUATION OF RADIO WAVES
Under this condition the attenuation due to rainfall should be directly
proportional to the path length. Thus the path length in figure 7.24 can
be shortened to correct for in.sufficient fading margin.
Bell System results indicate that for their microwave relay links in the
extreme southeastern region of the United States rainfall will limit 11
Gc/s radio systems having a 40 dB fading margin to path lengths of
approximately 10 to 15 mi, depending on the number of hops, if normal
reliability objectives are to be met. Path lengths of 20 to 30 mi should
be acceptable in the central area and paths as long as 35 mi should be
acceptable in the northwestern part of the country. However, in their
existing point-to-point radio relay systems, the paths average about 23
mi due to other considerations than those of propagation. It appears
that the 11 Gc/s systems will not be penalized unduly except in the
southeastern part of the United States.
An illustration of the correlation between the rainfall and path loss on
March 15, 1956, of the ]\Iobile study is presented in figure 7.26 in support
of the above conclusions.
CALCULATED
(RAINFALL DATA)
MEASURED
1200
TIME OF DAY
Figure 7.26. Correlation between rainfall and path loss, March 15, 1956.
(After Hathaway and Evans, 1959).
7.11. Attenuation by Hail
Ryde concluded that the attenuation caused by hail is one-hundredth
that caused by rain, that ice crystal clouds cause no sensible attenuations,
and that snow produces very small attenuation even at the excessive rate
of fall of 5 in. an hour. However, the scattering by spheres surrounded
by a concentric film of different dielectric constant does not give the same
effect that Ryde's results for dry particles would indicate [24, 27]. For
example, when one-tenth of the radius of an ice sphere of radius 0.2 cm
melts, the scattering of 10 cm radiation is approximately 90 percent of
the value which would be scattered by an all-water drop.
ATTENUATION BY FOG
303
At wavelengths of 1 and 3 cm with 7 = 0.126 (7 = 2a/X; a = radius
of drop) Kerker, Langleben, and Gunn [27] found that particles attained
total-attenuation cross sections corresponding to all-melted particles when
less than 10 percent of the ice particles were melted. When the melted
mass reached about 10 to 20 percent, the attenuation was about twice that
of a completely melted particle. These calculations show that the attenu-
ation in the melting of ice immediately under the °C [28] isotherm can
be substantially larger than in the snow region just above, and under some
circumstances, greater than in the rain below the melting level. Further
melting cannot lead to much further enhancement, apparently, and may
lead to a lessening of the reflectivity of the particle by bringing it to
sphericity or by breaking up of the particle. This effect, combined with
the fact that hail has greater terminal velocities than rain, gives rise to
the so-called "bright band" near the 0° isotherm.
7.12. Attenuation by Fog
The characteristic feature of a fog is the reduction in visibility. Visi-
bility is defined as the greatest distance in a given direction at which it
is just possible to see and identify with the unaided eye (a) in the day-
time, a prominent dark object against the sky at the horizon, and (b) at
night, a known, preferably unfocused moderately intense light source [29].
Although the visibility depends upon both drop size and number of
drops and not entirely upon the liquid-water content, yet, in practice,
the visibility is an approximation of the liquid-water content, and there-
fore, may be used to estimate radio-wave attenuation [28]. On the basis
of Ryde's work, Saxton and Hopkins [30] give the figures in table 7.11
for the attenuation in a fog or clouds at °C temperature. The attenua-
tion varies with the temperature because the dielectric constant of water
varies with temperature; therefore, at 15 and 25 °C the figures in table
7.11 should be multiplied by 0.6 and 0.4 respectively. It is immediately
noted that cloud or fog attenuation is an order of magnitude greater at
3.2 cm than at 10 cm. Nearly another order of magnitude increase
occurs between 3.2 cm and 1.25 cm.
Table 7.11. Attenuation caused by clouds or fog [SO]
Temperature=0 °C
Visibility
Attenuation (dB/km)
X = 1.25cm
X=3. 2 cm
X = 10cm
m
30
90
300
1.25
0.25
0.045
0.20
0.04
0.007
0.02
0.004
0.001
304 ATTENUATION OF RADIO WAVES
7.13. Thermal Noise Emitted by the Atmosphere
General laws of thermodynamics relate the absorption characteristics
of a medium to those of emission. Good absorbers of radiation are also
good emitters, and vice versa. Thus, in the microwave region, the atmos-
phere is also a good emitter, as well as a strong absorber, of radiation.
We may, therefore, describe quantitatively both emission and absorp-
tion by the same parameter; namely, the absorption coefficient.
The emission characteristics of any real body at a fixed frequency may
be compared to those of a blackbody at the same temperature. In the
microwave region, the noise intensity emitted by a blackbody is given by
the Rayleigh -Jeans law:
i(.) = 2kT[-J (7.8)
where \l/{v) = emitted blackbody flux density per unit frequency
V = frequency
T = absolute temperature, °K
c = the velocity of light, and
k = Boltzmann constant (1.38054 X 10-i« erg/°K)
The emission per unit length along an actual ray path may now be ex-
pressed as
Biu) = yiu) 4^ip) (7.9)
where 7(1') = attenuation per unit length. Remembering that the fraction
of energy absorbed in a path length, ds, is given by the optical depth,
dr = y{v)ds, we may obtain the differential equation for transmission of
radiation through the atmosphere:
^ = -/(.) + V(«^) (7.10)
where liu) is the flux density per unit frequency.
The solution to this radiative transfer equation is
liv) = i:lm{v) exp (- / "W)
+ / 4^{v) exp {- \ dr) dr (7.11)
THERMAL NOISE EMITTED BY ATMOSPHERE 305
where the summation extends over all discrete noise sources which may
be present, I m{v) is the unattenuated flux density transmitted from the
mth discrete source located at position r^, s is the point of reception of
energy, and the other symbols have their previous meaning. It should
be recognized that the above integrals extend over a ray path determined
by the refractive properties of the medium and cannot be evaluated
unless these refractive properties are known.
In analogy to the temperature dependence of the noise energy as by
the Rayleigh-Jeans law, we may, in the microwave region, relate the
intensity of radiation received from a particular direction, I{v), to an
equivalent temperature, T n{v), by the following relation
/(^) = ?^ (7.12)
or, from (7.8)
TM = ^TnAv) exp(- / "'rfr)
+ / T^expf-/ dr) dr. (7.13)
This equivalent temperature is called the thermal noise temperature.
It is apparent that the thermal noise temperature of the atmosphere,
as measured by an antenna, will depend explicitly upon the antenna angle
and the frequency, and implicitly upon the atmospheric conditions along
the ray path giving rise to attenuation and emission of energy. It seems
plausible, therefore, that one could exploit this dependence of thermal
noise on atmospheric conditions as a probe of atmospheric structure.
Thermal noise is equally important in communications receiving, since
it represents the lowest possible noise level that can be attained by an
antenna immersed in the atmosphere. This minimum noise level will,
of course, vary, depending on atmospheric conditions, the frequency, and
the antenna orientation. For example, in the microwave region, the
antenna noise temperature at vertical orientation may be as low as 1 °K,
and in a horizontal position, where more of the lower layers are "seen"
the noise temperature may be of the same order as the actual tempera-
ture of the lower atmosphere; i.e., around 280 °K. Figure 7.27 shows
sky temperature as frequency for various anteinia angles for mean atmos-
pheric conditions at Bismarck, N. Dak., during February 1940-43.
306
ATTENUATION OF RADIO WAVES
Figure 7.27. Thermal noise versus frequency for mean profile conditions
at Bismarck, N.Dak.
THERMAL NOISE EMITTED BY ATMOSPHERE
Table 7.12
307
Gas
ymax
Percent by
volume
at ground
7 at ground
Mc/s
12, 258. 17
(dB/km
1.9X10-1
(Oto l)X10-6
(dB/km)
(0-1.9)X10-'
12, 854. 54
8.7X10-1
(0-8. 7) X 10-7
23, 433. 42
1.2X10-1
(0-1. 2) X 10-'
SO2
24, 304. 96
2.3
(0-2.3)X10-6
25, 398. 22
2.1
(0-2. l)X10-«
29, 320. 36
3.3
(0-3.3)X10-«
44, 098. 62
5.2
(0-5.2)X10-6
52, 030. 60
9.5X10-1
(0-9. 5)X10-'
24, 274. 78
2.5
0.5X10-6
1.25X10-6
N2O
22, 274. 60
2.5
1.25X10-6
25, 121. 55
2.5
1.25X10-6
25, 123. 25
2.5
1.25X10-6
NO2
26, 289. 6
2.9
(0 to 2) X 10-8
(Oto 5. 8) XI 0-8
10, 247. 3
9. 5X10-2
Summer
(0 to. 07) X 10-6
Winter
(0to.02)X10-«
(0 to 6. 3) X 10-9
O3
11,075.9
9. 1X10-2
(0 to 6. 3) X 10-9
42, 832. 7
4.3X10-1
(0 to 2. 8) X 10-8
X
C2
UJ
X
LlI
_1
<
o
(f)
4
10
20
30
40
AUGUST. H,
FEBRUARY, H^
FEBRUARY. H,
AUGUST, H^
50
60
FREQUENCY IN Gc/S
Figure 7.28. Variations with frequency of the scale heights of the bi-exponential
absorption model at Verkhoyansk, U.S.S.R.
308 ATTENUATION OF RADIO WAVES
7.14. References
[1] Bussey, H. E. (Aug. 1950), Microwave attenuation statistics estimated from
rainfall and water vapor, Proc. IRE 38, No. 7, 781.
[2] Davidson, D., and A. Pote (Dec. 1955), Designing over-horizon communication
links. Electronics, 28, 126.
[3] Bean, B. R., and R. Abbott (May-Aug. 1957), Oxygen and water vapor absorp-
tion of radio waves in the atmosphere, Geofis. Pura Appl. 37, 127-144.
[4] Van Vleck, J. H. (Apr. 1947), Absorption of microwaves by oxygen, Phys. Rev.
71, 413-424.
[5] Van Vleck, J. H. (Apr. 1947), The absorption of microwaves by uncondensed
water vapor, Phys. Rev. 71, 425-433.
[6] Van Vleck, J. H. (1951), Theory of absorption by uncondensed gases. Book,
Propagation of Short Radio Waves, pp. 646-664 (McGraw-Hill Book Co., Inc.,
New York, N.Y.).
[7] Becker, G. B., and S. H. Autler (Sept. 1, 15, 1946), Water vapor absorption of
electromagnetic radiation in the centimeter wavelength range, Phys. Rev. 70,
Nos. 5 and 6, 300-307.
[8] Birnbaum, G., and A. A. Maryott (Sept. 15, 1955), Microwave absorption in
compressed oxygen, Phys. Rev. 99, 1886.
[9] Artman, J. O., and J. P. Gordon (Dec. 1954), Absorption of microwaves by oxygen
in the millimeter wavelength region, Phys. Rev. 96, No. 5, 1237-1245.
[10] Tinkham, M., and M. W. P. Strandberg (July 15, 1955), Line breadths in the
microwave magnetic resonance spectrum of oxygen, Phys. Rev. 99, No. 2,
537-539.
[11] Hill, R. M., and W. Gordy (Mar. 1954), Zeeman effect and line breadth studies
of the microwave lines of oxygen, Phys. Rev. 93, 1019.
[12] Artman, J. O. (1953), Absorption of microwaves by oxygen in the millimeter
wavelength region, Columbia Radiation Lab. Rept. (Columbia Univ. Press,
New York, N.Y.).
[13] Straiton, A. W., and C. W. Tolbert (May 1960), Anomalies in the absorption of
radio waves by atmospheric gases, Proc. IRE 48, No. 5, 898-903.
[14] Tolbert, C. W., and A. W. Straiton (Apr. 1957), Experimental measurement of
the absorption of millimeter radio waves over extended ranges, IRE Trans.
Ant. Prop. AP-5, No. 2, 239-241.
[15] Gunn, K. L. S., and T. W. R. East (Oct.-Dec. 1954), The microwave properties of
precipitation particles. Quart. J. Roy. Meteorol. Soc. 80, 522-545.
[16] Bean, B. R., and B. A. Cahoon (Sept. 1957), A note on the climate variation of
absolute humidity, Bull. Am. Meteorol. Soc. 28, No. 7, 395-398.
[17] Ratner, B. (1945), Upper air average values of temperature, pressure, and relative
humidity over the United States and Alaska (U.S. Weather Bureau).
[18] Donaldson, Ralph J., Jr. (June 1955), The measurement of cloud liquid-water
content by radar, J. Meteorol. 12, No. 3, 238-244.
[19] Weickmann, H. K., and H. J. aufm Kampe (June 1953), Physical properties of
cumulus clouds, J. Meteorol. 10 204-221.
[20] Battan, L. J. (1959), Radar Meteorology, Book, p. 43 (Univ. of Chicago Press,
Chicago, 111.).
[21] Ryde, J. W., and D. Ryde (1945), Attenuation of centimeter waves by rain, hail,
fog, and clouds (General Electric, Wembly, England).
[22] Laws, J. O., and D. A. Parsons (Apr. 1943), The relationship of raindrop size to
intensity. Trans. Am. Geophys. Union, 24th Annual Meeting, 452-460.
[23] Ryde, J. W. (1946), The attenuation and radar echoes produced at centimetre
wave-lengths by various meteorological phenomena. Meteorological Factors in
Radio Wave Propagation, pp. 169-188 (The Physical Society, London, England).
REFERENCES 309
[24] Hathaway, S. D., and H. W. Evans (Jan. 1959), Radio attenuation at 11 IcMc and
some implications affecting relay systems engineering. Bell Syst. Tech. J. 38,
No. 1.
[25] Burrows, C. R., and S. S. Atwood (1949), Radio wave propagation, Consolidated
Summary Technical Report of the Committee on Propagation, NDRC, p. 219
(Academic Press, Inc., New York, N.Y.).
[26] Mie, G. (1908), Beitrage zur Optik triiber Medien, speziell Kolloidaler Metal-
lasungen, Ann. Physik, 25, 377.
[27] Kerker, M., M. P. Langleben, and R. L. S. Gunn (Dec. 1951), Scattering of
microwaves by a melting spherical ice particle, J. Meteorol. 8, 424.
[28] Best, A. C. (1957), Physics in Meteorology (Pittman and Sons, London, England).
[29] Glossary of Meteorology (1959), Am. Meteorol. Soc. 3, 613.
[30] Saxton, J. A., and H. G. Hopkins (Jan. -Feb. 1951), Some adverse influences of
meteorological factors on marine navigational radar, Proc. IRE 98, Pt. Ill, 26.
[31] Zhevankin, S. A., and V. S. Troitskii (1959), Absorption of centimetre waves in
the atmosphere, Radioteknika i Elektronika 4, No. 1, 21-27.
[32] Button, E. J., and B. R. Bean (June 1965), The biexponential nature of tropo-
spheric gaseous absorption of radio waves, Radio Sci. J. Res. NBS 69D, No. 6,
885-892.
[33] Ghosh, S. N., and H. D. Edwards (Mar. 1956), Rotational frequencies and
absorption coefficients of atmospheric gases. Air Force Surveys in Geophysics
(Air Force Cambridge Research Center, ARDC, USAF).
Chapter 8. Applications of Tropospheric
Refraction and Refractive Index Models
8.1. Concerning the Bi-Exponential Nature of
the Tropospheric Radio Refractive Index
8.1.1. Introduction and Background
The recent explosive growth of space science and telecommunications
has spurred the development of new models of the tropospheric radio
refractive index to account for the systematic refraction of radio waves
and the calculation of theoretical radio field strengths at satellite heights.
The simple exponential model has been found to represent, to a first
approximation, the average refractive index structure within the first few
kilometers above the ground for the United States [1],^ France [2], and
Japan [3]. All of the above investigations have reported varying degrees
of departure of the atmosphere from this model and Misme [4] has en-
deavored to delineate the regions of the world where the exponential
model is most applicable, although subsequent analysis of several types
of data has shown this model to be more generally applicable than at first
sight and not unreasonable for use even in arctic and tropic locations [5].
If one considers that A^ is composed of a dry term,
and a wet term,
(8.1)
,^ = -^^g X 10'. (g^2)
then one may consider the height variation of each term separately. We
shall examine the possible advantages of a model of the form
N(z) = Do exp {-^} + Wo exp {-^} (8-3)
to describe the average decrease of A'^ with height, where Do and Wo are
the values of the dry and wet comi)onents at the earth's surface and Hd
* figures in brackets indicate the literature references on p. 373.
311
312 REFRACTION AND REFRACTIVE INDEX MODELS
and H^, are the scale heights of D and W, respectively. This particular
form has been found useful by Katz [6], in his derivation of the potential
refractive modulus, and by Zhevankin and Troitskii [7] in their treatment
of atmospheric absorption. It would be well for the reader to recall
that scale height, as used in this study, is merely the height at which the
value of the atmospheric property has decreased to 1/e of its surface value.
Typical values of Do, Wq, and A^o are listed for arctic, temperate, and
tropical locations in table 8.1. It is seen that the contribution of W to
the total value of A^ is nearly negligible in the arctic but becomes greater
as one passes from temperate to tropical climates. There is, of course,
generally an inverse correlation between the magnitude of D and W since,
at sea level, where P '^ 1,000 mbar, the low arctic temperature increases
the D term and, combined with low atmospheric water vapor capacity,
decreases the wet term. Conversely, the higher temi^eratures of the
temperate and tropical climates depress the D term and provide a greater
water vapor capacity with the result that W may have a sizable contribu-
tion to the total A^.
Table 8.1. Typical average values of the dry and wet components of N
station and climate
Do
Wo
No
Isachsen (78°50' N), arctic
Wasliington, D.C. (38°50' N), temperate
Canton Island (2°46' S), tropic
332.0
266.1
259.4
0.8
58.5
111.9
332.8
324.6
371.3
8.1.2. N Structure in the I.C.A.O. Atmosphere
One may examine A^ structure in a standard atmosphere as a guide to
its general distribution in the free atmosphere. On this basis the I.C.A.O.
standard atmosphere [8] (fig. 8. 1) was examined. The conditions specified
for this atmosphere are an approximately exponential pressure decrease
with respect to height and a linear temperature decrease from ground level
to the tropopause. It is evident, then, that in this atmosphere D de-
creases in an exponential fashion with height.
When these data are converted into refractive index and plotted on
semilogarithmic paper, as on figure 8.2. both D and W are seen to display
an approximately exi^onential distribution from the surface to the tropo-
pause. This conclusion is based upon the observation that the distribu-
tion is nearly linear as one would expect if one inverted the function
y = A exp i-h/c) (8.4)
N STRUCTURE IN I.C.A.O. ATMOSPHERE
313
50
40 -
MESO7PEAK
OZONOPAUSE
H = 47gpkm
400 800 1200 210 230 250 270 290
P(mbar) T(°K)
100 200 300
D = 77.6 Y
Figure 8.1. The U.S. extension to the I.C.A.O. Standard Atmosphere.
into
\n y = — h/c + In ^,
(8.5)
which is the equation of a straight Hne on semilogarithmic paper. The
exponential distribution of W with height in this atmosphere follows
naturally from the definition of constant relative humidity, since the
saturation vapor pressure, e^, is itself, to a first approximation, an ex-
ponential function of temperature. It is evident that the value of W can
significantly affect the surface values of A^ but has no appreciable effect
upon the value of A'' at the tropopause.
An examination of long term means from observations in the actual
atmosphere shows that this same general bi-exponential trend is observed
in practice, for temperate climates at least. Examples are given on
figures 8.3 and 8.4 for Bismarck, N.Dak., and Brownsville, Tex.
Bismarck is typical of the high, dry great plains region of North
America which is frequently subjected to strong intrusions of arctic air,
while Brownsville represents the humid periphery of the Gulf of Mexico.
Even in these very dissimilar climates one finds a strong tendency towards
a bi-exponential distribution of A^, particularly when W is large.
314
REFRACTION AND REFRACTIVE INDEX MODELS
1,000
700
500
N
RH = 100%
RH = 67 %
RH = 33%
RH =0% (DRY)
TROPOSPHERE
STRATOSPHERE
W(h),RH=IOO%
W(h),RH = 67 %
W(h),RH= 33%
4 6 8 10 12
H,gpkm
Figure 8.2. N distribution for the I.C.A.O. standard atmosphere.
14
8.1.3. Properties of the Dry and Wet Term Scale Heights
The average values of D and W versus height were determined for
22 U.S. radiosonde stations located about the country. The data used
were the published values of mean pressures, temperatures, and humidities
for the United States [9] which may be converted into mean values of the
refractive index with negligible error [10]. The method of least squares
was then used to determine the scale heights of the wet term, H^, and of
the dry term, Hd. Examination of these scale heights did not reveal any
simple method of predicting their geographic and seasonal behavior, other
than simply to map them. Such maps were prepared for the United States
DRY AND WET TERM SCALE HEIGHTS
315
for both winter (February) and summer (August). The immediate con-
clusion that one reaches from these maps, figures 8.5 and 8.8, is that H^
has a year round, country-wide average value of perhaps 2.5 km while
H d has an average value of about 9 km.
Since the scale height is the height at which the value of an atmos-
pheric property has decreased to \/e of its surface value it reflects the
degree of stratification of the property. For example, cold arctic air is
very stratified with very little vertical motion, with the result that its
density scale height would be expected to be low. By contrast, tropical
maritime air that has moved over land is characteristically unstable with
convective activity thoroughly mixing the original moist surface air
N
1,000
700
500
300
200
100
70
50
30
20
10
7
5
3
2
2 4 6 8 10 12 14
ALTITUDE ABOVE M.S.L. IN km
Figure 8.3. N distribution for Bismarck, N.Dak.
Note that altitude rather than geopotential height is used here to facilitate the eventual calculation of radio-
ray bending through actual atmospheric layers.
! 1 i 1 1
1
ELEVATION =505m
—
:
- AUGUST 7^"''"'^^
A^ ^D(h)
- \
\
\ ^AUGUST
-
\ \-^W(h)
\ *^
\ \
- FEBRUARY^A. \
-
316
REFRACTION AND REFRACTIVE INDEX MODELS
1,000
700
500
N
ELEVATION = 7m
- ^FEBRUARY
2 4 6 8 10 12 14
ALTITUDE ABOVE M.S. L. IN km
Figure 8.4. N distribution for Brownsville, Tex.
throughout the entire troposphere, with the result that the density scale
height is relatively larger than in the case of arctic air. The dry term
scale height on figures 8.5 and 8.7 show a distinct tendency to be larger
during the warm summer months when the atmosphere is well mixed to
great heights. Consequently D decreases slowly with height.
A slight geographic pattern is observed in the Hd maps: the coastal
regions display somewhat higher values than the inland regions. The
north-south direction of the isopleths along the west coast on the February
Hd map definitely reflects the uniform onshore advection of low-density
maritime air. By contrast, the east coast shows an east-west isopleth
pattern, the high values in Florida reflecting the well-mixed nature of sub-
tropical air, and the lower values in New England inaicating the presence
DRY AND WET TERM SCALE HEIGHTS
317
Figure 8.5. Dry term scale height, Hd, in kilometers for February.
Figure 8.6. Wet term scale height, Hw, m kilometers for February.
318 REFRACTION AND REFRACTIVE INDEX MODELS
Figure 8.7. Dry term scale height, Hd, in kilometers for August.
Figure 8.8. Wet term scale height, Hw, in kilometers for August.
DRY AND WET TERM SCALE HEIGHTS 319
of the more dense and stratified continental air that customarily flows
offshore during the winter months. The same pattern is repeated on the
summertime map along the west coast but is less i^ronounced on the east
coast due to the combination of more uniform heating and also onshore
advection of maritime air |)roduced by the circulation pattern of the
Bermuda high-pressure area. The high value of H d = 10.5 km observed
in the southwest during the summer appears to be due to the intense heat-
ing with resultant convective mixing to great heights so common in that
desert area. A somewhat opposite pattern is evidenced by the //„, maps.
For example, the coastal areas generally have the lowest values and thus
reflect the characteristic strong humidity stratification of maritime air.
The smaller humidity gradients of the inland regions produce somewhat
larger scale heights for that area. The summer Hy, map is quite sur-
prising in that very little variation is shown, perhaps indicating uniform
vertical convection of the available moisture at all locations throughout
the country. The strong convection indicated in the southwest on the
summer H d map is again reflected by the high value of //,„ = 3.0 km for
that same area.
It is quite evident from figure 8.9 that within the troi)osphere, /i < 10
km, the bi-exponential model has a lower rms error for the common,
near-zero angles of departure used in tropospheric propagation of radio
waves. Both models yield about a 12 percent error in determining
T for dt) — Q and h = W km. At ^o = 100 mrad, however, the percentage
error decreases to 4 percent for the bi-exponential model and 7 percent for
the exponential reference atmosphere. The rather marked errors of the
single exponential model at 10 km simply reflect that that model is
deliberately fitted to the average A'^ structure over the first few kilometers
with the result that this model systematically departs from the average
atmospheres in the vicinity of the tropo pause. This is particularly
apparent at the higher values of ^o where the integral tends to become a
function of the limits of integration. That is, using the theorem of the
mean for integrals.
cot d dn ^ -cot da dn ^ [N^ - N] 10-" cot ^o (8.9)
under the assumption that cot 6 may be replaced by cot ^o over the interval
of integration. For do < 20 mrad this assumption introduces less than
a 10 percent error for the interval < /i < 10 km. It is ai)parent from
(8.9) that the error in j^redicting r for large do is then simj)ly a matter of
how closely the model approaches the true value of n in the atmos}ihere.
At ^0 near zero, however, the integral for r is very heavily weighted to-
wards the effect of n gradients near the earth's surface [1]. Since the
values of do commonly used in tropospheric propagation are as near as
320
REFRACTION AND REFRACTIVE INDEX MODELS
practically possible to zero, and both models show comparable rms errors
for do = 0, one concludes that there is no clear advantage to the bi-
exponential model for this application. This conclusion is furthered by
the fact that charts of Hd, Hy,, Do, and Wo are not available, while to use
the single exponential model one need only to refer to existing regional
or worldwide maps [10, 12].
4.0
(T
or
o
(r
(r
UJ
CD
Q
LxJ
CD
^ 0.3
0.4
=
EXP. REF
I -EXP.
9„ = ?0mr
9n-- 50 mr
EXP REF
BI-EXP
n
0n=IOOmr
-EXP REF
-BI-EXP
0,5
0,4
0.3
0.2
0.1
0.20
0.16
0.12
0.08
0.04
cr
o
cr
cr
UJ
LU
CD
CD
cr
3 10 15 70 3 10 15 70
HEIGHT IN KILOMETERS
Figure 8.9. Rool-mean-square errors of predicting bending for both the bi-exponential
refractive index model and the C.R.P.L. (single) exponential atmosphere.
Height is used here to indicate the actual thickness of atmosphere traversed by the radio ray
REFRACTION IN BI-EXPONENTIAL MODEL 321
8.1.4. Refraction in the Bi-Exponential Model
The test of a model of atmospheric refractive index is the degree to
which it represents the average A^ structure of the atmosphere. A
further critical test is the degree to which the refraction, or bending, of a
radio ray is represented by this atmosphere. The bending is given as
the angular change of a radio ray as it passes from rii to n-i in a spherically
stratified atmosphere and is adequately represented by
Ti,2 = - / cot ddn (8.6)
where 6 is the local elevation angle and is determined at any point from
Snell's law,
nr cos 6 = rioro cos ^o, (8.7)
where r is the radial distance from the center of the earth and zero sub-
script denotes the initial conditions. It is customary to evaluate (8.6)
numerically [11] since the integral is intractable for all but the most
simple models of n versus r. The bending was obtained for the mean A^
profiles for one-half of the 22 U.S. weather stations mentioned earlier.
The values of r predicted by the bi-exponential model for these same
stations were obtained by preparing U.S. maps of Hd and H,j, from the
other half of our data, selecting values oi Hd and Hu, for the test stations,
calculating the bending and obtaining rms differences between these
values and those obtained from the mean A'' profiles. These rms differ-
ences are shown on figure 8.9. Also, for comparison, the rms errors
obtained from the CRPL exponential atmosphere are shown. This
atmosphere, based upon a single exponential curve passing from the
surface value, N s, to the value at 1 km, A^i, is founded upon the expression
A^i = A^, - 7.32 exp {0.005577 A^^} (8.8)
which has been found to be applicable in the United States [1].
An obvious advantage of the bi-exponential model is that the scale
heights do reflect the physical properties of the atmosphere in a much
clearer way than does the single exponential model.
8.1.5. Extension to Other Regions
The present study is based upon data from the continental United
States and one wonders if the same approach might be of utility in other
regions. As a brief check, the D and W term scale heights were deter-
mined for conditions typical of the long arctic night (Isachsen, Northwest
322 REFRACTION AND REFRACTIVE INDEX MODELS
Territory, Canada, for February) and for humid tropical areas (Canlon
Island, South Pacific, February) from 5-yr means of A'^ versus height
and are listed in table 8.2.
The extreme meteorological differences of these two locations are quite
evident. The Ha = 9.4 km and 7/^, = 2.0 km at Canton Island indicate
a warm atmosphere with a strong humidity gradient, while the value
of H d = 6.3 km at Isachsen indicates very stratified air with high surface
density and a strong density decrease with height. The value of //„, =
6.5 km at Isachsen reflects a very low humidity gradient; in fact, at no
point in the troposphere does W exceed 3 A^ units for this location and
season.
Table 8.2. Hd and Yi^ifor arctic and tropical locations
Station
Hrf
Hu.
Canton Island
km
9.4
6.3
km
2.0
Isachsen, N.W.T.,
Canada..
6.5
The value of //„, = 2.6 km reported for the characteristic altitude of
water vapor for the middle belt of the U.S.S.R. [7] also appears to be in
agreement with the conclusions of the present study.
8.2. Effect of Atmospheric Horizontal
Inhomogeneity Upon Ray Tracing
8.2.1. Introduction and Background
It is common in ray tracing studies to assume that the refractive index
of the atmosphere is spherically stratified with respect to the surface of
the earth. Thus, the effect of refractive index changes in the horizontal
direction is normally not considered, although Wong [13] has considered
the effect of mathematically smooth horizontal changes in airborne propa-
gation problems.
Neglecting the effect of horizontal gradients seems quite reasonable in
the tropospheric because of the relatively slow horizontal change of
refractive index in contrast to the rapid decrease with height. In fact,
examination of climatic data indicates that one must compare sea-level
stations located 500 km from each other on the earth's surface in order to
observe a difference in refractive index values which would be comparable
to that obtained by taking any one of these locations and comparing its
surface value with the refractive index 1 km above the location. Although
the assumption of small horizontal changes of the refractive index appears
CANTERBURY 323
to be true in the average or climatic sense, there are many special cases,
such as frontal zones and land-sea breeze effects, where one would expect
the refractive index to change abruptly within the 80-odd kilometers of
horizontal distance traversed by a tangential ray passing through the
first kilometer in height.
It is these latter variations that are investigated in this section. Two
cases of marked horizontal change of refractive index conditions were
studied, one which occurred over the Canterbury Plain in New Zealand,
and the other at Cape Kennedy, F)a. Although these particular sites
were chosen for several reasons, such as land-to-sea paths and subtrojMcal
location (where marked changes in refraction conditions are common), the
major consideration was that detailed aircraft and ground meteorological
observations were available for prolonged periods.
These detailed measurements allow a quantitative evaluation of the
error apt to be incurred by assuming that the refractive index is hori-
zontally stratified. The procedure used was to determine the refractive
index structure vertically over the transmitter and assume that this same
structure vertically described the atmosphere everywhere. Rays were
then traced through this horizontally laminated atmosphere. These ray
paths were then compared with those obtained by the stcp-by-step ray
tracing through the detailed convolutions of refractive index structure in
the two cases under study.
In the sections that follow we will discuss the two cases chosen for
study, the methods of calculation used to evaluate refraction effects, and
the degree of confidence to which standard prediction methods may be
used under conditions of horizontal inhomogeneity.
8.2.2. Canterbury
The Canterbury data were compiled by a radio-meteorological team
working from September 1946, through November 1947, on the South
Island of New Zealand under the leadership of R. S. Unwin [14]. This
report proved invaluable in this investigation, as it was very carefully
prepared, giving minute details of the experiment on a day-to-day basis.
Anson aircraft and a trawler were used for meteorological measurements
over the sea, and three mobile sounding trucks for observations on land.
The trucks and the trawler carried wired sonde equipment, whereby
elements for measuring temperature and humidity up to a height of from
150 m to 600 m (depending on wind conditions) were elevated by means
of balloons or kites. Standard meteorological instruments provided a
continuous record of wind, surface i)ressure, temi)erature, and humidity
at stations at the coast and 14 km and 38 km inland. The headtiuarters
of the [)roject were at Ashburton Aerodrome, and the observations ex-
tended out to sea on a fine })erpendicular to the coastline of Canterbury
324 REFRACTION AND REFRACTIVE INDEX MODELS
Plain. Aircraft were equipped with a wet-bulb and dry-bulb psychrom-
eter, mounted on the })ortside above the wing. Readings were taken
three or four times on each horizontal flight leg of 2 or 3 min duration.
Special lag and airspeed corrections were applied, resulting in accuracy
of ±0.1 °C. It was found that, under the variety of conditions in which
observations were made, the aircraft flights were more or less parallel to
the surface isobars; hence, the sea-level pressure as recorded at the beach
site was considered to hold over the whole track covered by the aircraft.
The relationship used for calculating the pressure, P, in millibars at a
height h in feet was:
P(h) = Po - h/30
where Po is the surface pressure. This approximation (determined by
averaging the effect of the temperature and humidity distributions on
pressure in a column of air) resulted in a maximum error in the refrac-
tivity of 0.5 percent at 900 m. Radiosonde ascents at Hokitika on the
west coast of South Island and Paraparaumuo and Auckland on North
Island were used to supplement the aircraft measurements, particularly
in the altitude levels above 1 km.
The observations, diagrams, and meteorological records were studied,
and a profile of unusually heterogeneous nature was chosen. The
synoptic situation for the morning of November 5, 1947, was selected, as
it revealed a surface-ducting gradient near the coast with an elevated layer
about 100 km off shore. A cross section of the area from Ashburton to a
point 200 km offshore was plotted with all available data, and isopleths of
modified refractive index, AI, were drawn to intervals of 2.5 units
M = N -^ {Keh)lO', (8.10)
where Ke = (15.70) (10-8)/m, and A^ is as defined in chapter 1, (1.20). A
simplified version of the lower portion of this cross section with the corres-
ponding M curves is accompanied by a sketch of the general location of
the experiment in figure 8.10. Some smoothing was necessary, particu-
larly near the sea surface and in those areas where aircraft slant ascents
and descents caused lag errors in altimeter readings and in temperature
and humidity elements. Isopleths over land were plotted above surface
rather than above sea level, with an additional adjustment in the scale
ratios of height and distance in an attempt to simplify the reading of
values from the diagram.
8.2.3. Cape Kennedy
The second area studied was the Cape Kennedy to Nassau path for the
period of April 24 to May 8, 1957. This material was supplied by the
Wave Propagation Branch of Naval Research Laboratories and the
CANTERBURY
325
METERS
900-1—
ISOPLETHS OF MODIFIED REFRACTIVE INDEX
4IO-X-rr4JO
0^--^ ^^- - :r_i~ -400
FEET
5000
METERS
300+--
200- -
20 20
—LAND SEA—
—LAND SEA
60 80 100 120 140
RANGE FROM COAST (KILOMETERS)
20 10 20 40
M CURVES
80 100
120 140 160
100
340
i
\k
t
W
T
I
w
V
340 340 340 340 340 340 340 340 340
HORIZONTAL SCALE= ONE DIVISION -10 M UNITS
340 340
800
«00
y1440o
V--200
-lU-0
340
NOTE: VERTICAL SCALE CHANGE ABOVE 300 KM
(ADAPTED FROM CAHTEBBURy PROJECT)
-NEW ZEALAND-
X DENOTES RAOB SITES
LAND OVER 1500 FEET
36 S
40 S
44 S
Figure 8.10. Isopleths and curves of refractive index for November 5, 1947, Canterbury,
with a map locating sources of meteorological data.
(Courtesy of the Canterbury project).
326 REFRACTION AND REFRACTIVE INDEX MODELS
University of Florida. The particular case chosen for study was the
meteorological profile of May 7, 1957 (2000 L.T.) due to its heterogeneous
nature, showing a well-defined elevated layer at about 1 ,500 m. Fourteen
refractometer soundings from aircraft measurements taken at various
locations along the 487-km path (fig. 8.11) and six refractive index
profiles (deduced from radiosonde ascents from Cape Kennedy, Grand
Bahama Island, and Eleuthera Island) were read in order to plot a cross
section of the atmosphere which would represent as closely as possible
the actual refractive conditions at that time. Unfortunately, the data
near the surface (up to 300 m) were quite sparse compared to those
recorded in the Canterbury Project, and calibration and lag errors had
not been noted as carefully in this preliminary report; therefore, some
interpolation and considerable smoothing of refractive index values were
necessary when drawing isopleths.
8.2.4. Ray Bending
The classic expression for the angular change, t, or the bending of a
ray passing from a point where the refractive index is Wi to a second point
where the refractive index is rii is given in chapter 3 as
r,^, = - Z'^cot^, (8.11)
n
where 6 is the local elevation angle. Equation (8.11) was evaluated by
use of
ni,2
where
- gi + 02
The value of 6 at each point was determined from Snell's law:
niTi cos di = 712^2 cos 02 = constant, (8.13)
where r is the radial distance from the center of the earth and is given
by a + h, where a represents the radius of the earth and h the altitude
of the point under consideration. For simplicity one may rewrite
(8.13) as
(1 + A^i X 10-«) (a + hi) cos Oi
= (1 + A^2 X 10-6) (q + /j^) cos 02. (8.14)
RAY BENDING
327
ISOPLETHS OP REPRACTIVE INDEX
CAPE CANAVERAL
220 280'^ 340
DISTANCE IN KILOMETERS
REPRACTOMETER PLIGHT PATH
ELEUTHER A ISLAND
■p =X=Sj^ SAN SALVADOR 'X^
X DENOTES RAOB SITES
Figure 8.11. Isopleths of refractive index and map of refractometer flight path for
May 7, 1967, Cape Kennedy to Nassau.
328 REFRACTION AND REFRACTIVE INDEX MODELS
Then, when 6 is small, one may expand (8.14) [as in chapter 3, (3.50) to
(3.58)] and obtain the convenient expression:
1^2 ^ 2(h, -h,) _ 2(^^ _ ^^) ^ jo-ejy^
(8.15)
where all values of 6 are in milliradians.
After obtaining r by use of (8.12) and (8.15), one may determine the
distance, d, along the earth's surface that the ray has traveled from:
rfi,2 = a [ri,o + (62 - di)]. (8.16)
Thus by successive application of the above formulas, one may trace
the progress of the radio wave as it traverses its curved path through
the atmosphere. Normally the use of these equations is quite straight-
forward. When considering horizontal changes in n, however, one must
satisfy these equations by iterative methods. In the present application,
since n had to be determined by graphical methods, it was felt to be
sufficient to assume a constant distance increment of 250 to 500 m, solve
for appropriate height increment from
Ah = Ad tan di
1 + '
a.
(8.17)
graphically determine N for the point di + Ad, hi + Ah, and then deter-
mine do and Ti,2.
This latter type of ray tracing was done for various rays of initial
elevation angles between 261.8 mrad (15°) and 10 mrad (^0.5°). The
calculations were not carried to smaller elevation angles, since this type
of ray tracing is not valid within surface ducts for initial elevation angles
below the angle of penetration [16].
8.2.5. Comparisons
Although both of the calculated ray paths consisted of an oversea
itinerary with coastal transmission sites, they are quite different in other
aspects. Canterbury Plain is located southeast of the 10,000-ft chain
of the Southern New Zealand Alps at a latitude of 44°S (the equatorward
edge of the westerly belt of winds in November). Cape Kennedy is
located on a sea-level peninsula at 28°N (the poleward edge of the north-
east trade circulation in May). While the Canterbury profile showed
superrefractive tendencies, the Cape Kennedy profile illustrated sub-
refraction at the surface counterbalanced by an elevated trade wind
inversion layer, indicating that the total bending values of Canterbury
COMPARISONS 329
would be higher than normal, while the Cajje Kennedy example would
have values near or lower than normal.
These differences are illustrated by figures 8.12 and 8.13 where the
bending, r, in milliradians is plotted versus altitude in kilometers. The
effect of horizontal changes is most pronounced for rays with initial eleva-
tion angles of 10 mrad. On these figures the term "vertical" ray is used
to designate the ray path through the horizontally homogeneous n struc-
ture determined from the refractive index vertically over the station.
The term "horizontal" ray designates the ray path through the complex
actual n structure. It is quite evident that a consistent difference in
bending of about 1 mrad exists between the "vertical" and "horizontal"
rays at Canterbury above 1 km for ^o = 10 mrad. This would be expected
since the vertical M profile (fig. 8.10) at the beach (our hypothetical
transmitter site) is nearly normal in gradient while as little as 10 km to
sea a duct exists, thus indicating a near maximum difference between the
"horizontal" and the "vertical" rays at any initial elevation angle small
enough to be affected by the duct. This is in contrast, however, with
the case of Cape Kennedy where, except for the region of the elevated
duct centered at about 1,500 m, the "vertical" and "horizontal" rays are
in c^uite close agreement. These two examples illustrate that horizontal
variations must be near the surface to be most effective. The importance
of the altitude of the variation is due to the fact that refraction effects
are very heavily weighted toward the initial layers [16].
Also shown on figures 8.12 and 8.13 are the values of the bending which
would be predicted from the Central Radio Propagation Laboratory
corrected exponential reference atmosphere [1]. The values shown are
obtained from the value of A'' at the transmitter site as corrected by the
vertical gradient over the first 100 m. It is noted that, for do = 10 mrad
at Canterbury, the value of bending predicted by the model is in essential
agreement with the "vertical ray" bending but under estimates the
"horizontal ray" (which has the largest variation of n with horizontal
distance) by about 1.25 mrad. For Cape Kennedy at ^o = 10 mrad,
the model atmosphere overestimates the bending by about 1.25 mrad for
altitudes in excess of 2 km. It should be emphasized that, although the
model exponential atmosphere appears to represent the average of the
two specific cases studied, the departure from this average arises from
quite different causes in each case. The differences in the Canterbury
case arise from the marked effect of horizontal variation of n as is indi-
cated by the agreement of the vertical ray bending with the model atmos-
phere. The disagreement in the Cape Kennedy case is due to the
presence of a very shallow surface layer of nearly normal gradient topped
by a strong subrefractive layer; therefore, it represents a shortcoming of
the model rather than an effect of horizontal changes of n.
330
REFRACTION AND REFRACTIVE INDEX MODELS
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bJ
I-
UJ
9 6
UJ
Q
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< 2
(J) 8
UJ
q 6
Ld 4
Q
Z)
< 2
j
7^
/ /
/
/ I
/ /
/
o
o
E
V
x,lf
4
/
00
io
00
/
Hi
1
/if y
$ /
1
/
/ //
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1
1
1
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^
/
Y / /
/
y
^^00 Ho
,^^\0 mrad "
1
Vert.
1
/
^
l^.
^'
17.4 mrac
/? -^ ^
(^
i J^^ y
J c
o^.r, „^. _ 1
i
fe:^^^
r^
Corrected Exponential
-2 2 4 6 8 10
BENDING IN MILLIRADIANS
Figure 8.12. Canterbury, to 10 km, altitude versus ray bending.
'/
/
//
/
/
/
'
T3
/
;
/
/
//
1
1
1
ti
/ /
e
/
^1
!«>
-V^
t-7
/
^1
s
1 1^-
1^
' .(
/
^
i 1
/
1
/
1
/ /
/
!
i 1
/
I
j
1
f 1
/
i
1
1
//
/ /
/
I
Jl
/
/ /
/
1
It f
/
/ /
/
1
jl /
/
' /
/
!
^"'/
/
/
/
J
i
/''
/
00 Hor
Vert.
j^ /
■'y'
10 mrad '-
111 //
J
^
^^
y-y^ y'
17.4 mrad <^
°
/>-^
"-^^s
^^
52.4 mrad -^ »
81.8 mrad" °
<s
'"'"^
'^
Corrected Exponential
2 4 6
BENDING IN MILLIRADIANS
Figure 8.13. Cape Kennedy, to 10 km, altitude versus ray bending.
EXTENSION TO OTHER REGIONS
331
The preceding analysis of bending throws the refractive differences in
each case into sharp rehef. The effect of refraction, of course, is to vary
the ray path. Figures 8.14 and 8.15 show the ray paths corresponding
to the bendings of figures 8.12 and 8.13. Note that for Canterbury at
^0 = mrad, the effect of the horizontal variation of n is to produce a
difference in estimation of about 1 km in height or 20 km in ground dis-
tance at 300 km from what would be obtained from considering the verti-
cal n profile as a representation of the entire path. The effect of the
subrefractive layer at Cape Kennedy is not so large, but it does cause an
overestimation of the ground distance by about 5 km and an underesti-
mation of the height by less than one quarter of a kilometer at a ground
distance of 300 km by assuming that the vertical profile may be used
throughout the entire ray path.
8.2.6. Extension to Other Regions
It should be pointed out that the ducting case at Canterbury represents
an extreme refraction condition and is not necessarily typical of conditions
observed in other regions or, indeed, at Canterbury. The Canterbury
project was purposely restricted to a study of ducting conditions with the
result that less than 20 percent of the total observations for the fifteen
Corrected exponential reference
atmosphere
Path of Canterbury vertical ray
Path of Canterbury horizontal ray
120 160 200 240
DISTANCE IN KILOMETERS
Figure 8.14. Canterbury, to 10 km, attitude of ray versus distance.
332
REFRACTION AND REFRACTIVE INDEX MODELS
LJ
O
_J
3
1
/
/
^
i>/
f.
^-^
^1
V
.->
Ko-^
r
4i?
r
>^
j
/
■ >^
■/'
/
/
^
^
r ^
/
/
y^
^^ r„..l.t„^
'
y^ ^
Y^
atm
■ Path
/
/ \^^\
^sphere
of Canaveral vertical ray -
of Canaveral horizontal ray
^
o
5 Pnth
^
^
1 1 1 1
40
240
320
360
DISTANCE IN KILOMETERS
Figure 8.15. Ca-pe Kennedy, to 10 km, altitude of ray versiis distance.
months are reported. Therefore, because one of the more extreme cases
is represented by the November 5 example, one might conclude that much
less than 20 percent of the observations would show the same degree of
horizontal n change as the profile studied.
If one further hypothesizes that the greatest horizontal n change would
be associated with ducting conditions, then the percentage incidence of
ducts as evaluated from radiosonde observations, listed for various
stations in table 8.3 would indicate that the effects of horizontal changes
of n sufficient to cause variations in the ray path as large as those of the
present study would be observed less than 15 percent of the time, regard-
less of geographic location.
The probable importance of subrefractive layers upon the prediction
of refraction effects has emerged as a secondary result of the present study.
Although subrefraction is normally neglected, it is potentially a very im-
portant refractive factor for distances of, say, less than 40 km. Even
though the percentage occurrence of subrefractive layers can be as large
as 6 percent (see table 8.4), this effect is frequently offset by the con-
current occurrence of an adjacent superrefractive layer, as is illustrated
by the Cape Kennedy example.
CONCLUSIONS 333
Table 8.3. Percentage occurrence of surface ducts during the years 1952 to 1956
Station
Percent incidence
February
May
August
November
Fairbanks, Alaska
Columbia, Mo
Washington, D.C.
9.4
0.7
0.7
10.0
0.7
0.4
2.5
4.8
9.2
3.5
0.4
8.4
4.3
12.4
8.5
6.2
1.3
1.4
Canton Island
11.5
Miami, Fla
2.7
Table 8.4.
Percentage occurrence of surface subrefractive layers during the years
1952 to 1956
Station
Percent incidence
February
May
August
November
Fairbanks, Alaska
Columbia, Mo
Washington, D.C
0.0
0.0
0.9
0.0
0.7
0.0
1.6
2.2
0.0
0.3
1.2
0.6
5.8
0.0
0.9
0.4
4.0
2.7
Canton Island
0.3
Miami, Fla
0.7
8.2.7. Conclusions
The conclusions of the present study could be considerably modified
by the analysis of many more examples, although it is evident that hori-
zontal variation of n near the earth's surface produces the most marked
deviations from the ray paths obtained by assuming horizontal stratifica-
tion of n. The effect of horizontal changes occurring more than a kilom-
eter above the surface appear from our present examples, to have little
effect. Further, the effects of horizontal changes appear to be most
pronounced in the presence of surface ducts and at small elevation angles.
The tentative conclusion is reached that the effect of horizontal n change
is normally small, since ducting will occur less than 15 percent of the time.
8.3. Comparison of Observed Atmospheric Radio Re-
fraction Effects With Values Predicted Through
the Use of Surface Weather Observations
8.3.1. Introduction
The atmospheric radio refraction effects considered in this section are
of two general types: errors in measuring distance by means of timing
the transit of radio signals between two points, known as radio range
errors, and errors in estimating the elevation angle of a target by means
of measuring the angle of arrival of radio signals from the target, known
334 REFRACTION AND REFRACTIVE INDEX MODELS
as elevation angle errors. Many methods have been proposed to take
into account these refraction effects for the purpose of improving measure-
ments by removing systematic bias. One of these involves the use of
the surface value of the radio refractivity, A''^, a quantity which can be
measured directly with a microwave refractometer, or calculated from
the ordinary meteorological variables of temperature, pressure, and
humidity, to predict values of either range error or elevation angle error;
this method has been shown theoretically to be useful, with the accuracy
increasing with increasing initial elevation angle [17, 18, 19]. It is the
purpose of the present note to compare recent experimental determina-
tions of atmospheric refractive effects with values estimated theoretically
from surface meteorological conditions.
8.3.2. Theory
The operation of a radio tracking system depends on the measurement,
in some manner, of radio signals received from the target. The radio
signals are transmitted in the form of radio waves which travel from the
target to the tracking system. The form of these radio waves is distorted
by the presence of the earth's atmosphere. Since solutions of the wave
equation are extremely difficult to obtain for the case of general atmos-
pheric propagation over a spherical earth, it is common practice to
evaluate refraction effects by means of ray tracing, a process which is
based on the use of Snell's law.
One of the two types of refraction errors considered in this appendix
is the elevation angle error, c, which is the difference between the apparent
direction to a target, as indicated by the angle of arrival of a normal to
the radio wave front, and the true direction. This error is primarily a
function of the refraction, or bending, of the radio ray. For targets
beyond the atmosphere, the two quantities are asymptotically equal (with
increasing range). . The values of e and r at any point on the ray path
obey the following inequality:
r/2 ^ e ^ T.
Recalling that (chapter 3) the bending of a radio ray may be expressed
by an equation of the form
T = a -\-hNs (8.18)
where a and b would be functions of the initial elevation angle of the ray,
do, and the height (or range) along the ray path at which the bending is to
be calculated. Such an assumption can be checked by examining the
behavior of values of r, ray traced for a number of observed height profiles
of radio refractive index, plotted against the corresponding values of N s-
THEORY 335
Such a plot is shown in figure 8.16 for a small initial elevation angle,
50 mrad (about 3°), and a "target" height beyond the atmosphere, 70 km.
The family of A^ profiles used in ray tracing this sample of bending values
is referred to as the CRPL Standard Sample. ^ It can be seen from inspec-
tion of figure 8.16 that the assumption of linearity expressed in (8.18) is
justified for this case. A similar conclusion can be reached from examina-
tion of data for other cases, including low target heights and elevation
angles down to zero degrees, although for these extremes the degree of
correlation between t and N s is not as marked as that shown in figure 8.16.
The other refraction variable treated in this section is the radio range
error, ARe, which is here defined as being that error incurred in measuring
the distance between two points by means of timing the transit of radio
signals between the points, and assuming that the velocity of propagation
is equal to that of free space. For the case of a radio ray, this error is
composed of two parts: the difference between the curved length of the ray
path, called the geometric range, Rg, and the true slant range, Ro; and
the discrepancy caused by the lowered velocity of propagation in a refrac-
tive medium. The geometric range is given by
Rg = / CSC 6 dh,
and the apparent, or radio, range by
J n
Jo
Thus the total radio range error, ARe — Re — Ro, is given by
fht
ARe = I n CSC 6 dh — Ro,
Jo
Re ^ I n esc d dh.
or
ARe ^ 10 ^ f N CSC ddh -\- I esc d dh - Ro. ,^ ,^.
Jo (o.iyj
^Meaning explained in section 8.3.3.
336 REFRACTION AND REFRACTIVE INDEX MODELS
Table 8.5. Typical and extreme values of range errors for targets beyond the atmosphere
Typical N,=
= 320
Extreme Ns=
=400
Maximum
percent
00
ARg
ARw
ARe
ARg
A/e.v
ARe
ARg/ARe
meters
10
100
110
60
165
225
27
20 mrad
2.5
62.5
65
4.5
73
77.5
6
50 mrad
0.7
38.1
38.8
1.0
43
44
2.3
100 mrad
0.14
22.26
22.4
0.2
24.8
25
0.8
200 mrad
0.02
11.9
11.9
0.03
13.0
13.0
0.23
500 mrad
0.001
5.01
5.01
0.002
5.50
5.50
0.04
The first term on the right-hand side of (8.19) is the "velocity" or
"refractivity" error, ARn', the last two terms represent the geometric
range error, ARg, which is the difference in length between the straight
path, Ro, and the curved ray path, Rg. Table 8.5 gives some typical and
extreme values of range errors ray traced for observed A'^ profiles.
7.0
T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r
-T=-0,685476 + 0.01804535 Ng (mrad)
J I I L
J I I L.
Figure 8.16. Total refraction at do equals 50 mrad, h equals 70 km, for the CRPL
standard sample.
THEORY 337
From table 8.5 it can be seen that the geometric range error, ARg, does
not represent a significant portion of the total range error except at very-
small initial elevation angles, between zero and about 3°. This being so,
the behavior of the total range error will be primarily a function of the
first integral in (8.19) for elevation angles greater than about 3°. The
integral expression
ARe ^ 10~' / ' N csed dh,
may be rewritten as
ARe ^ CSC ^0 X 10"'
N dh
1 - 2 sin' ( — T-^ ) + cot 00 sin {d - do)
or
ARe ^ CSC do N dh
+ Z (-l)''-"^ N
cot 00 sin (0 - do) -2 sin'
^)]
dh, (8.20)
for sin d < 2 sin do, < d < 7r/2. This expression is analogous to that
derived for the case of ray bending, (3.6) of chapter 3, and, similarly, the
integral series on the right-hand side of (8.20) contributed only 3 percent
or less to the value of ARe for do larger than about 10°. From (8.20) one
would thus suspect that the radio range error might be well e.stimated as
a linear function of the integral of N with respect to height. In treating
this integral, it is informative to note that any given N(h) profile may be
"broken up" into three primary components:
N(h) = N'{Ns, h) + N"ih + hs) + 8N(h)
where A'^' is that part of the profile which can best be expressed as a
function of A^^ and height, N" is a standard distribution of refractivity
with resjiect to altitude above mean sea level (h -\- hs) which is independ-
ent of N s, especially above the tropopause, and 5N represents a random
component of the profile which cannot in general be accounted for
a priori.
338 REFRACTION AND REFRACTIVE INDEX MODELS
The A'^' component is generally effective over the first few kilometers,
while above 6 or 7 km altitude, the A'^" component forms the bulk of the
profile [1]. Thus the integral of the A^ profile with respect to height may
be written as:
Ndh = I N' (Ns, h)dh-\- / N" {h + h,)dh + / 8N{h) dh,
Jo Jh, Jo
or.
Ndh ^ F, (Ns, hd + F2 (K, ht) + 6F (ht)
where 8F is the random contribution to the integral. For any particular
ht then
Ndh = Fy (Ns) + F2 (hs) + dF
or
where
N dh = F, (hs = 0) + Fi (A^^) - F3 (hs) + 8F (8.21)
F3 = / N"(h + hs) dh, and F2{hs = 0) is a constant.
It was found empirically, from integrated N{h) profiles, that
N dh^a + biNs - hihs ^ S.E. (8.22)
The analogy between (8.21) and (8.22) is plain (the standard error of
estimate of (8.22), SE, represents the standard deviation, SF, of (8.21)).
The results of such an empirical study are shown in figure 8.17 for the
CRPL Standard A'^ Profile Sample, for ht beyond the atmosphere.
For any particular application of (8.22) at a single location, the term
62/1 s will be absorbed into the constant a, since hs does not vary. How-
ever the introduction of this term is necessary to explain the station
THEORY
339
2,5
CO
or
I^^ iO-y°° N(h) dh (m
I^= 1.4588 + 0.0029611 Ng (m)
r = 0.982
S.E.= 0.0368nn
I I I
250
350
Figure 8.17. Integrated refractive index -profiles for the CRPL Standard Sample.
elevation dependence of integrated N{h) profiles when taken from a
sample containing stations at widely differing elevations, such as the
Standard Sample.
It is thus apparent that radio range errors, at least at the higher eleva-
tion angles, are primarily a linear function of N s- That this is also true
at comparatively low angles is shown in figure 8.18, for ^o = 50 mrad
(about 3°) for the same profile sample. The reader should especially note
the similarity of the distributions of the points about the regression lines
between figures 8.17 and 8.18, showing that the range errors at about 3°
are still primarily a function of the integral of A'^ with respect to height,
or the range error at 90°.
It has thus been demonstrated that, theoretically, it should be possible
to estimate both the angle of refraction of radio rays and errors in radio
range measurements from measurement of the refractive index at the
surface of the earth. This should be true for targets in or beyond the
atmosphere, at elevation angles down to, and possibly lower than, 3°.
In addition, if the behavior of refractive index profiles is similar in differ-
ent parts of the world, it should be possible to specify "universal" values
340
REFRACTION AND REFRACTIVE INDEX MODELS
45
en
40 -
35
30
1 1 1 1 1 1
1 1 1 1 1 1 I 1 1 1 1 1
1
_ AR'e = l8.2968 + Q063l05 Ns^m)
REGRESSION OF
ARg ON Ns AND hg
'^\ °V
'^-
- AR'e= ARg + 1.962
hs ^„ ^^° °°
-
n POINTS FOR DENVER, COLO., AND
°
ELY, NEV. ; SHOWN ALSO AS:
t
Q
• ADJUSTED FOR ELEVATION
-
1 1 1 J 1 1
DEPENDENCE (ARg)
1
1 , 1
1 1 1 1 1 1 1 1 1
210
250
350
400
Figure 8.18. Total range error at do equals 50 mrad, h equals 70 km, for the CRPL
standard sample.
of the coefficients in (8.18) and (8.22) and to predict these values in ad-
vance by analysis of a large heterogeneous sample of refractive index
profiles. In the succeeding subsections it will be shown how this has
been done, and a comparison will be made between the results so derived
and the results of some measurements over actual radio paths.
8.3.3. The CRPL Standard Atmospheric Radio
Refractive Index Profile Sample
In the preceding section it was shown that, theoretically, it should be
possible to estimate either radar elevation angle errors or radio range
errors at any particular location by means of a system of linear equations
in N s, where the coefficients are functions of the target position. The
target position can be specified by either the apparent elevation angle and
target height, or the apparent range and target height (or as a third possi-
bility, the apparent range and elevation angle), each having advantages
in different situations [19]. The equations recommended are
= ai (00, hi) + bi {do, ht) Ns ± S.E.i {do, he),
(8.23)
REFRACTIVE INDEX PROFILE SAMPLE 341
and
ARe = a. (Re, ht) + 62 [Re, ht) Ns ± S.E.2 (Re, ht), (8.24)
where e is the elevation angle error, AR e is the radio range error, ^0 is the
apparent elevation angle, hi is the target height, Re is the apparent radio
range, and S.E. is the standard error of estimate about the regression
line of € or ARe on N s- Values of the coefficients may be obtained by
performing linear regressions of e or ARe, as ray traced for an appropriate
sample of radio refractive index profiles, ujjon A^^, for a large number of
target positions. As a byproduct of these calculations, one also obtains,
for each target position, a value of the residual error (the standard error of
estimate) to be expected for the particular type of profile sample used.
In order to obtain a general set of eciuations to be useful under arbi-
trary conditions of location, climate, and weather, a large sample of
N profiles has been assembled which is believed to be representative of
both mean climatic and geographic trends and the larger synoptic varia-
tions which may be encountered. This was done by choosing 13 radio-
sonde stations representative of the major geographic and climatic types
of the world, and then choosing from each station six A'' profiles of particu-
lar types, two of which are typical of the extremes of monthly mean
conditions for that location, and the other four of which are typical of
some of the variations which are found at that location [18]. The result
is a sample of 77 N profiles,^ which has been found over a period of years
to be a sound cross section of general refractive conditions and has thus
been named the CRPL Standard Atmospheric Radio Refractive Index
Profile Sample, hereafter referred to as the CRPL Standard Sample.
Although the locations chosen for this sample are heavily weighted towards
the United States, it has been found that the general behavior of the
refractive index structure as inferred from the standard sample is typical
of conditions experienced in most parts of the world [5].
The remainder of this section will be devoted to some comimrisons of
observed radio refraction data with the predictions supplied by the
CRPL Standard Sample, as derived from the linear regressions men-
tioned above.
Since the refraction measurements re[3orted here consist of samples
taken at particular locations over comparatively short i)eriods of time,
they should provide a test for the general set of coefficients derived from
the Standard Sample; not only is the general theoretical api)roach tested
against measured values, but the measurements coming from places of
more or less homogeneous nature, they provide a check as to whether 01
not coefficients derived for a large heterogeneous sample of data are
applicable also to individual places and times; i.e., they should reveal
One of the types could not be found for one of the stations used.
342 REFRACTION AND REFRACTIVE INDEX MODELS
how much of the observed correlation of the heterogeneous sample is
derived from correlation between "classes" of data (in the statistical
sense). For a more thorough treatment of the CRPL Standard Sample
and the associated regression coefficients for range error and elevation
angle error, the reader is referred to Bean and Thayer [19].
For the Standard Sample, the standard error of estimate is equal to
the standard prediction error within ±1 percent over the range of A^s from
200 to 470, and will be used interchangeably with the latter.
8.3.4. Comparison With Independent Data
Before turning to an examination of the experimental refraction data
and the degree of success realized in applying the theoretical prediction
model to those data, it seems appropriate to examine the accuracy of the
prediction model when ai^plied to some independent theoretical (i.e., ray-
traced) data. For this purpose, four check stations were selected which
were not only independent in the sense of not having been included in
the original 13-station Standard Sample, but were from locations widely
differing from the region of selection of the original sample. It was de-
cided to select one station representative of an arctic type climate, one
temperate, one tropical, and one from a "problem" climate area.
Amundsen-Scott station at the South Pole (lat. 90°S) was chosen as the
arctic type ; this station was expected to present the most rigorous test of
the prediction model (as based on the Standard Sample) that could be
obtained anywhere in the world. In the first place the extreme arctic-
continental climate, with almost no water-vapor contribution to the re-
fractive index and the nearly incessant temperature inversion, is more
alien to the Standard Sample than any other type ; in the second place the
station elevation is 2800 m, which is 900 m in excess of the highest station
(Ely, Nev., 1908 m) included in the Standard Sample. These two effects
were expected to augment each other as regards refraction.
Dakar, Senegal, on the western coast of Africa, was selected as a
"problem" climate station; an inverse relationship exists there between
N s and AN (the A'-gradient over the first kilometer above the surface).
A Congo basis station, Bangui, in what was French Equatorial Africa,
was selected as the tropical location, and Moscow, U.S.S.R., was selected
as the temperate location.
In order to combine brevity with comprehensiveness, ray tracings were
done of the total refraction (bending at 70 km target height) at two
elevation angles, 20 mrad and 100 mrad, for six profiles from each loca-
tion. The six profiles were selected as representing roughly the range of
Ns in winter (February), summer (August), and spring-fall (May and
November), two profiles being selected from February, two from August,
and one from each of May and November. The 20-mrad elevation angle
COMPARISON WITH INDEPENDENT DATA
343
— I 1 1 I 1
AMUNDSEN -SCOTT STATION
SOUTH POLE, ANTARCTICA
LAT = 90° S
LONG. - N A.
ELEV = 2800 m
OBSERVED FIT-
•=-5.89»tt0520 Ns
S E =«006mtad^^ |8„ = 20n
^STANDARD PREDICTION
ERROR LIMITS
(67% CONFIDENCE)
^STANDARD SAMPLE
T = -OI40t000963N5
(PREDICTION ERROR: lOOl r
DAKAR, SENEGAL, AFRICA
LAT= 14° 43' N
LONG =17° 14' W
ELEV. = 40 m
-OBSERVED FIT
l56>003l6Ns
-'\ SE = «033iT
!£l STANDARD SAMPLE
r'-287>00358Ns
^STANDARD PREDICTION
ERROR LIMITS
(67% CONFIDENCE)
^—STANDARD SAMPLE
T = -QI40. 0,00963 Ns
(PREDICTION ERROR = '0 01
200 210 220 230 240 250 260 2?0
320 330 340 350 360 370
MOSCOW, USSR
LAT =55° 49' N
L0NG = 37°37'E
ELEV = 186m
20 mrad| ,,---C-i:
^OBSERVED FIT
r=-4l9.00390Ns
SE ='0 ISmrad
STANDARD SAMPLE
T=-0l40>000963Ns
(PREDICTION ERROR. >0 01 r
BANGUI, CENTRAL AFRICAN REPUBLIC
LAT = 4° 23' N
LONG = 18° 34' E,
ELEV =385 m
"■STANDARD PREDICTION
ERROR LIMITS
(67% CONFIDENCE)
OBSERVED FIT
= -6 l7-00457Ns
S E = t0 34rnrod
STANDARD SAMPLE
T = -0 I40'000963N5
(PREDICTION ERROR = 'OOImrod)
290 300 310 320 330 340 350 360
310 320 330 340 350 360 370 3«0
Figure 8.19. Comparison of predicted refraction at h equals 70 km for Oqs of 20 mrad
and 100 mrad from regressions of the CRPL standard sample and ray-traced values
from four independent locations.
344 REFRACTION AND REFRACTIVE INDEX MODELS
was selected as representing roughly the lower limit of elevation angles
for which the bending is expected to be strongly correlated with N s (say
r > 0.9), while at 100 mrad (about 6°) the correlation is expected to be
extremely high (say r > 0.99) and the refraction should be reasonably
free of random profile effects.
The results of the ray tracings and the comparison with predicted
values are shown in figure 8.19. As expected, the results from the South
Pole seem to depart significantly from the predicted values, at least for
the 20-mrad elevation angle. At the 100-mrad elevation angle some of
the calculated points lie more than one standard deviation from the pre-
dicted line (the theoretical prediction error is too small to show on the
graph clearly) ; however, in all four cases the differences are less than 50
^trad, a figure which as shall be seen may represent the limit of accuracy
obtainable from the atmosphere in actual practice. At angles over 100
mrad the errors would be smaller; in fact they should tend to decrease in
inverse proportion to the square of the initial elevation angle, as indeed
they do between 20 and 100 mrad.
A conclusion which may be drawn from the above results is that any
regions where the prediction model based on the Standard Sample would
not be expected to provide the theoretical accuracy are probably regions
of climatic extremes, and at least for the case of angular errors the effects
will be negligible for elevation angles of a few degrees or more. As an
interesting aside it can be noted that apparently the Antarctic may be a
desirable area for tracking systems location, at least with respect to
atmospheric refraction effects, since (most likely because of the lack of
substantial water vapor and the relatively homogeneous conditions) the
prediction error for ^o = 20 mrad in figure 8.19 is only about one-fifth as
large as for temperate climates, indicating a possibly more stable atmos-
phere (even 90 percent confidence limits for the SE in figure 8.19 yield a
value less than half of the theoretical temperate value of ±0.286 mrad).
8.3.5. Comparison With Experimental Results
Before comparing the theoretical and experimental results, it is appro-
priate at this point to examine what one would expect to observe on the
basis of propagation theory. In the case of angular errors it is expected
that i^ropagation through the real, turbulent atmosphere will produce
random variations in the shape of the incoming wavefront, so that meas-
urements made with systems in which the receiving antenna is alined with
the incoming signal will have random variations introduced in addition to
the ordinary refraction effects. Since these variations will probably not
be a function of elevation angle to any great extent, this implies that the
residual variance in predicting the elevation angle errors will probably
always be greater than predicted from theoretical (static) considerations,
and that there will probably be some minimum value of this variance for
COMPARISON WITH EXPERIMENTAL RESULTS 345
very large elevation angles. Thus, in some cases, the residual errors will
probably not decrease steadily with increasing elevation angle, but will
tend to flatten out at some point and assume a more or less constant value
above that point. These effects will be complicated in comparing one set
of data with another by such things as differences in the location or time
of day or season in which data are taken, and instrumental effects such as
aperture averaging.
The case of range errors is more straightforward. The effects of
turbulent atmospheric inhomogeneities are expected to average out over
regions of abnormally high or low density, when considering the transit
time of particular points on the wave front. Hence the effect on the
residual range errors is expected to be small, and the observed values are
expected to compare rather well with the predicted (theoretical )values.
Turning first to the comparison of observed and predicted elevation
angle errors, figure 8.20 shows some data on the mean refraction of 1.85-
cm radio waves received from the sun, a target at essentially infinite range
so that the elevation angle error is identical with the total angular bending
of the radio ray, r. The data shown in figure 8.20 were obtained by
tracking the sun with a precise radio sextant developed by the Collins
Radio Company, and were collected in August through December of 1959
at Cedar Rapids, Iowa [20]. These data represent essentially instan-
taneous measurements. The mean of all observations at each elevation
angle is plotted for elevation angles ranging from 2 to 65°, and the mean
value of N s associated with each point is about 332; the curve for the
mean bending of the CRPL Standard Sample corresponds to the mean
value of A^s of 334.6 for that sample and hence the data should be com-
parable. The standard deviation "wings" refer to the standard deviation
of the individual "instantaneous" data, not to the standard error of
estimate of the mean value. The close agreement observed for elevation
angles between 2 and 35° constitutes not only a confirmation of the use-
fulness of the Standard Sample, but also a verification of the accuracy
of ray-tracing theory in estimating radio wave refraction in the actual,
and thus heterogeneous, atmosphere. The standard deviation of the
Collins data (shown on the lower part of fig. 8.20) is generally lower than
for the standard sample, but this is to be exjjected in view of the larger
range of climatic variation contained in the CRPL Standard Profile
Sample. The apparent discrepancies in the measurements made at eleva-
tion angles over 40° are apparently due to some slight inaccuracies in the
calibration procedure used on the radio sextant during the period of data
acquisition.'* In fact, the data shown in figure 8.20 are almost precisely
'' The data for the highest elevation angles in figure 8.20 were necessarily collected
during the early part of the period when the sun was higher in the sky. In a private
communication, Anway states that the mean A'^s applicable to the data at 60° to 65°
was 358 rather than 332; this difference would account for about one-third of the
discrepancies noted, reducing the residual bias to a maximum of about 40 Mi'ad.
346
REFRACTION AND REFRACTIVE INDEX MODELS
ELEVATION ANGLE, 9^, DEGREES
26° 40" 55° 65°
Figure 8.20. Comparison of measured total atmospheric refraction of 1.85 cm radio
waves at Cedar Rapids, Iowa, with values predicted from Ns.
what one would expect to observe if all of the measured values of refrac-
tion were increased by a systematic calibration error of about 50 /xrad
over their correct values. The standard deviations in figure 8.20 tend
to flatten out at high elevation angles, an effect which is to be expected
theoretically as pointed out previously. At any rate, the largest differ-
ence between the observed data and the predicted curve in figure 8.20
at elevation angles over 30°, is only about 50 /urad or 10 sec of arc (the
angular diameter of the planet Mars at its average distance from the
earth is 10 sec of arc, an angle not discernible to the naked eye). Al-
though this discrepancy might be significant militarily, it is only about
0.5 percent of the diameter of the target sun and is probably near the
limit of accuracy of the equipment used.
COMPARISON WITH EXPERIMENTAL RESULTS
347
Figure 8.21 shows the results of the specific measurements reported by
Anway for the radio sextant for all cases at an elevation angle of 8° zt
0.09°; each point represents an "instantaneous" reading. The soHd hne
represents the Hnear regression of the measured refraction data on the
values of A''s; the dashed line shows the predicted linear relationship
derived from least squares fits to the CRPL Standard Sample ray traced
refraction data. The mean bias between the two lines is about 40 jurad,
interestingly close to, and in the same direction as, the apparent cali-
bration error noted in the mean refraction data at high elevation angles.
The standard error of estimate is considerably higher than predicted ; how-
ever, the rms uncertainty of ±0.052°, or ±0.91 mrad, in the apparent
elevation angle would be sufficient by itself to increase the standard error
of estimate to about ±0.017 mrad, which is 4 times larger than the pre-
dicted value. It is not known how much of the total standard error of
±0.12 mrad is due to measurement errors as opposed to unforeseen
fluctuations in actual atmospheric refraction.
<
<
2.5
<
Li_
LJ
1.5
;LEAST SQUARES FIT
-0.216 + 0.00754 Ns (mrad)
0.87, S.E. = 0.12 mrad $ Ij
\^
MEASURED REFRACTION OF 1.85 cm RADIO WAVES
AT 8° ± 0.052° ELEVATION ANGLE , WITH
UNCERTAINTY WINGS DUE TO INEXACT
ELEVATION ANGLE : rdX.] „„ = 0.0163
^LINEAR REGRESSION OF C.R.P.L. STANDARD
N- PROFILE SAMPLE; PREDICTED VALUES;
T = -0.058 + 0.00698 Ns (mrad)
r = 0.999, S.E. = 0.004 mrad
\ \ \ \
290 300
310
320
330
340
350 360 370 380 390
Figure 8.2 L 1.85 cm radio refraction at an elevation angle of 8 degrees, at Cedar Rapids,
Iowa.
(After Anway, 1961).
348
REFRACTION AND REFRACTIVE INDEX MODELS
T3
O
^ \ ^ \ ^ \
CAPE KENNEDY, FLA
MEAN APPARENT ELEVATION ANGLE =0.7 mrad
RANGE = 17. 1 km
1EAN TARGET HEIGHT = l3 7m
Figure 8.22. Elevation angle fluctuations from phase differences taken across a 24-ft
vertical baseline, at Cape Kennedy, Fla.
Figure 8.22 shows some results of measurements taken at Cape
Kennedy, Fla., on November 1-3, 1959, at a very low elevation angle,
about 0.7 mrad or 0.04°. These are "instantaneous" measurements,
taken at half-hourly intervals, of the phase difference fluctuations between
the signals from a beacon as they arrived at the upper and lower terminals
of a vertical 24-ft baseline, thus being very closely equivalent to a meas-
urement of the fluctuations in the angle of arrival of the wave front at the
centerpoint of the baseline (the altitude difference between this point and
the target beacon is referred to as the "mean" target height). Since only
the fluctuations and not the total phase differences were measured, only
the slope and scatter of the elevation angle errors as a function of the
observed N s data can be compared with the predicted values from the
CRPL Standard Sample. The zero point on the graph is set by the pre-
dicted mean value for the sample. The correlation coefficient is, as ex-
pected, only 0.57. In this case the scatter of the observed data is well
inside the limits of the standard error of estimate of the regression for the
standard sample, even at this very small elevation angle where horizontal
changes in the A'^ profile can exert a large effect on elevation angle errors.
COMPARISON WITH EXPERIMENTAL RESULTS
349
Figure 8.23 shows the results of a comparison between predictions of
elevation angle errors estimated from the CRPL Standard Sample and
some measurements made with a 6-cm radar at Tularosa Basin, N. Mex.
[21]. Each point represents the mean of five "instantaneous" readings
made at 1-min intervals over a period of 4 min. The standard deviation
of each five-reading group averaged 0.16 mrad, and the maximum range
in any one group was 0.58 mrad. The radio energy was propagated over
a 45-mi path at a mean apparent elevation angle of 18 mrad; the target
was a beacon located on a mountain peak 5610.5 ft higher than the desert
floor where the radar was located. The data in figure 8. 18 show that even
for this rather extreme case, where the degree of correlation between N s
and € is expected to be only 0.4, agreement is obtained between:
(a) the predicted and observed mean refraction,
(b) the observed and predicted slopes of the e versus N s relation, and
(c) the observed and predicted residual errors of predicting e from N s
alone.
O
a:
o
cr
or
<
O
h-
<
>
LaJ
I I I \
TARGET HEIGHT=I7I km
RANGE -72.5 krn
MEAN APPARENT
ELEVATION ANGLE = 19.0 mrad
STANDARD ERROR
h OF ESTIMATE
\ \ \
PREDICTED FROM
ORP.L, STANDARD SAMPLE
6p=0OI02Ns-l,50
SE = + 0.608mrad
OBSERVED REGRESSION:
eQ=O.OI03Ns-l.65
r=0 427
STANDARD ERROR
OF ESTIMATE
230 240 250 260 270 280 290 300 310 320 330
Ns
Figure 8.23. Measured refraction of C-band radar at Tidarosa Basin, N .Mex.
350
REFRACTION AND REFRACTIVE INDEX MODELS
C/5
ttr
LlI
1.5
TARGET HEIGHT =3.046 km
RANGE =25, 1 km
ELEVATION ANGLE = 7°
STANDARD ERROR
OF ESTIMATE
J I \ \ L
340
350 360 370
Ng-PUUNENE AIRPORT U.S.W.B.
380
Figure 8.24. Range error fluctuations observed on Maui path.
The small discrepancy between the intercepts (i.e., between the mean
refraction) of the observed and predicted e versus A'^ « lines may be perhaps
attributed to, for example, antenna lobe pattern distortion caused by
differential refraction, or defocusing [22].
The remaining data which are examined were of necessity taken in
such a manner as to have a rather high degree of autocorrelation (trends).
Such data are not as suitable for confirming the accuracy of a regression
prediction process as are independent data. A discussion of this is in-
cluded at the end of this section.
Turning to examination of radio range errors, figure 8.24 shows the
results of some measurements of apparent radio range fluctuations over a
25-km path on the island of Maui, Hawaii, on November 9-11, 1956 [23].
These measurements were made at 1-hr intervals, and are essentially
"instantaneous" values. The target beacon was situated on the summit
of Mount Haleakala at an elevation of 10,025 ft, while the "ground"
station was near Puunene Airport at an elevation of 104 ft, thus yielding
a target height of 3.046 km, in a region of critical target heights for pre-
diction of radio range errors in tropical climates [19]. The measured
range fluctuations (absolute errors not measured) are plotted against
values of N s taken at about the same time (mostly 15 to 20 min later) by
COMPARISON WITH EXPERIMENTAL RESULTS 351
U.S. Weather Bureau personnel at the Puunene Airport weather station.
The agreement between observation and prediction is fairly good, espe-
cially when one considers that only 32 of the 86 points lie outside of the
predicted standard error of estimate limits, while chance would indicate
that 29 points would exceed these limits. Also, it should be kept in mind
that in this case, as for all except the Collins data, the target beacon is
located on the surface of the earth, whereas the predictions from the
CRPL Standard Profile Sample are derived for targets in the free atmos-
phere; there is undoubtedly some bias introduced in this way.
As a part of a continuing investigation into the atmospheric limitations
imposed on electronic distance measuring equipment, some measure-
ments have been made recently by the Tropospheric Physics Section,
NBS, of both range errors and range difference errors (across a phase-
differencing baseline) over a propagation path near Boulder, Colo.
Figures 8.25 and 8.26 are based on some of the preliminary results of
these measurements [24]. Figure 8.25 shows the results of measurements
of the fluctuations in apparent range, made at half-hour intervals on
May 9-11, 1961, over a 15.5-km path between a transmitting beacon
r
LINEAR REGRESSION
r =0.86
SLOPE : 0,865 ppm/N
PREDICTED SLOPE, 0879ppm/N
THE STANDARD ERROR OF ESTIMATE
OF EITHER LINEAR FIT IS 5.4ppm,
THE PREDICTED VALUE = 5.8ppm.
E
Q.
a.
Figure 8.25. Range error fluctuations observed over the Boulder Creek-Green Mountain
path, Colo.
3-day run, half-hourly readings, range = 15.5 km, target height 088 m, approx elevation angle 26° May
9-12, 1961
352 REFRACTION AND REFRACTIVE INDEX MODELS
1
1
1 1
120
-
-
o
Q
o
H
Q
^^
„ °
, °
D-
^^ °
a>
o
3
no
-
^"°i^^^
„
-
i
„ o°
T^^W^ °
°
o
>==-
'*?*^''S2<.
^^-^^ OS
I
°
8
* o° °
°^-C^^^^° ^^OBSERVED SLOPE
z
<
100
~
° CO
PREDICTED SLOPE— -^^^^^^^^.^
~
tn
^-~~^^
~'
BOTH LINES HAVE ^
^
E
£
°
SE = 3.5mm, r =0.625
^^
<
90
1
1
I 1
-
N,-AT TWO ANTENNAS
Figure 8.26. Range difference fluctuations observed over a 460-m in-line baseline,
Boulder Creek-Green Mountain path, Colo.
4.15-hr run, 460 meter in-line baseline, target height = 688 m apparent range 15.5 km, May 9-11, 1961
on Green Mountain at an elevation of 2242 m and a receiving antenna
located near Boulder Creek at an elevation of 1554 m, the true target
height thus being 688 m.
The apparent range fluctuations, expressed in parts per million of the
15.5-km path length (with an arbitrary zero since the total range was not
measured), are plotted as a function of the surface value of the refractive
index taken at a point quite close to the lower terminal. Quite good
agreement is seen between the simple linear regression of the observed
AT^e values on N s and the predicted linear relationship obtained from
the CRPL Standard Sample. Note that both hues have statistically
equal standard errors of estimate with respect to the observed data.
Figure 8.26 shows the results of the range difference measurements
made over a 460-m baseline essentially in line with the transmission path,
where the second antenna was farther from the target beacon than the
primary antenna. Here the range difference fluctuations (again with an
arbitrary zero) have been plotted as a function of the mean value of N s
measured at each end of the baseline. The zero point on the graph is set
by the predicted mean of the sample. In this case there seems to be some
discrepancy between the regression of the data and the predicted slope;
DISCUSSION OF RESULTS 353
however, note that the standard errors of estimate for the two hnes are,
to two significant figures, equal, indicating that the difference in the
slopes is probably statistically insignificant.
There are some data points in figure 8.26 having a rather large devia-
tion from the regression lines. Statistical theory (using the "Student"
^-distribution for 84 deg of freedom) shows that, if the data points are
drawn from a normally distributed population, there should be only one
point having a deviation of more than ±9 mm from the observed regres-
sion line. There are in fact five points in figure 8.26, four above and
one below the line. If these five points are thrown out, on the grounds
that they weight too heavily the extremes of the distribution of data points
(this is especially true when using least squares regression), and the regres-
sion is then redone using the remaining 81 data points, the resulting value
of the slope is —0.385 mm/A'^-unit with r = 0.77 compared to the pre-
dicted slope of —0.381 mm/A'^-unit, a rather close agreement.
8.3.6. Discussion of Results
As a summary of the results of the experimental versus theoretical
comparisons given in the preceding section, a statistical analysis has been
run on the significance of the differences between the slopes of the ob-
served and predicted regression lines. In order to make the tests more
stringent, it was assumed that the slopes derived from the Standard Sam-
ple should be taken to be the slopes of the population regression lines (/3),
thus yielding an estimate of the significance of the departure of the ob-
served slope from the assumed population value.
A value of t was first calculated for each case using the relation [25]
_ I b - ^0 I V2(x. - xY
where b is the observed slope, ^o the assumed population, or theoretical,
slope, X refers to the independent variable in each regression, N s, SE is
the standard error of estimate, and ij_2 is the value of t for j — 2 deg of
freedom. From tj-2, confidence limits for /3 at the 100(1— a) percent
level can be calculated from [25]
lj^,aSE < ^ < ^ ^ tj^2,.SE ^g_26)
VS(Xi - x)2 \/2(Xi -
The probability that the observed value b would have fallen outside of
these limits by chance is a. Many statisticians consider a value of tj^o
354 REFRACTION AND REFRACTIVE INDEX MODELS
falling below the 100a = 5 percent level to be not significant, between the
5 percent and 1 percent levels to be of questionable significance, and over
the 1 percent level to be significant [25]. An observed slope b falling
Vi:(xi - xY
would thus be taken to represent a significant departure from the value
/So, and would thus imply the possibilities
(a) j3o does not represent /3, or
(b) h represents the regression of data from a population different than
that used in determining ^So, or
(c) both.
Before making the significance tests, however, the value of j, the num.-
ber of independent observations going into the determination of h, must
be known. In general, data of the type presented here are more or less
highly autocorrelated, and hence not all independent. The data pre-
sented here, with the possible exception of the Collins data and the
Tularosa Basin data for which the calculations could not be performed,
have autocorrelation coefficients Vk, for lag k {k = 1, 2, 3 units of time be-
tween successive measurements) that can be approximately described by
and for this type of data the effective number of pieces of independent
data, j, is given by [26]
■fr
— r
J = n
+ r'J
(8.27)
For the data treated here weighted mean values of r' were calculated
from
r. + ^rl-^9rl-+-^.^kM-
"^ - 1 + 4 + 9+-.. A;2 ' ^'^■^''^
where k was the largest lag for which the autocorrelation coefficient was
calculated, usually 4 or 5. No special justification is offered for the use
of (8.28) other than the obvious fact that Vk is to be approximated by
the kth. power of r', and hence a function of k would seem to be the most
logical weighting function to use; the use of k"^ as a weighting function
seemed to give the best overall fit to the series of r^ encountered from
these data.
DISCUSSION OF RESULTS
355
Table 8.6 shows the results of the significance tests on the slopes of the
various experimental and theoretical (predicted) regression lines. The
number of pieces of data is shown in the first column, the observed slope
b and theoretical slope /3 in the second and third columns, the autocorrela-
tion coefficient for lag of one time unit in the fourth column, and the
weighted mean r' as defined in (8.28) in the fifth column. In column 6
the effective number of independent pieces of data, j, is shown, while in
column 7 the value of tj^o is shown for the difference between b and /S.
The next column shows the value of ^_,_2, 0.5, the value for the 50 percent
significance level for j — 2 degrees of freedom.
Only one of the t values turns out to be significant at the 50 percent
level, which means that there was a better than even chance that such
differences would have occurred by chance in the other cases. In the case
of the Colhns data at ^0 = 8°, the value of t = 1.01 would not be significant
at the 25 percent level; the value ^46 = 1.01 corresponds to a = 0.34, or
a 34 percent chance that the observed deviation |6 — /3| is of a random
nature, and thus not significant.
Table 8.6. Experimental versus theoretical slopes
Is [6-/3] Sig-
e VS N,
n
6
Tl
r'
i
tj-2
t(0. 50,
>-2)
nificant at
the a =50
percent level?
Collins data 8°.. .
48
0. 00754
0. 00698
(48)
1.01
0.68
Yes
Tularosa Basin_..
161
0. 0103
0. 0102
(161)
0.031
0.676
No
Cape Canaveral..
86
0. 01356
0.00648
.870
.860
6.5
0.708
0.73
No
ARe VS N,
Maui data
86
0. 02833
0.01610
.974
.950
2.2
3.56
7.6
No
Boulder Creek-
Green Mt
155
0.865
0.879
.944
.950
4.0
0.32
0,82
No
A(Ai?) VS Ns
Boulder Creek-
Green Mt
86
-0. 344
-0. 381
.957
.946
2.4
0.60
2.0
No
From the point of view of a statistician, the results of these tests are
such that no significance can be attached to any of the apparent discrep-
ancies between theory and observation, and given reason to believe that
the values of /3 are theoretically sound, one could say that the results are
significantly positive in nature. The significance of the differences be-
tween the predicted and observed slopes of the regression lines for ^0 =
20 mrad for the independent data check of subsection 8.3.4 were tested
using the same method as the preceding tests, except that the six observa-
tions in each case were assumed to be independent. The results are
summarized in table 8.7 and confirm the general use of the Standard
Sample for do > 20 mrad.
356
REFRACTION AND REFRACTIVE INDEX MODELS
From the experimental data which are available at the present time
it may be concluded that:
(a) Radio range and elevation angle errors can be predicted from the
surface value of the radio refractive index, and the accuracy obtained will
be generally commensurate with the estimates of residual errors made
from theoretical ray-tracing considerations.
(b) The functional dependence of either angular refraction or range
errors on the surface value of the refractive index as derived from the
CRPL standard A'' profile sample may be applied to arbitrary locations
or climates without noticeable decrease in accuracy over that obtained
with a sample from the location under consideration.
(c) The effects of horizontal inhomogeneities of the refractive index,
which certainly must have been prevalent over the transmission paths for
which experimental data have been presented, do not appear to introduce
any bias or additional residual variance into the values of observed re-
fraction variables over those j^redicted from surface observations.
Table 8.7. Comparison of slopes for independent check
Predicted slope at 0o=2O mrad: 0. 358 mrad/iVs
Observed
slope
Difference
6-/3o
t
100a
Station
V^(N,
-NsV-
Significance
Amundsen-Scott
Dakar
Bangui
0. 0520
.0316
.0457
.0390
+0. 0162
-. 0042
+.0099
+.0032
37.9
56.5
47.2
44.3
10.2
0.72
1.37
1.18
<0. 1 percent
52 percent
25 percent
31 percent
very high
none
very low
Moscow
very low
-Nsr-
S. E.
8.4. Correction of Atmospheric Refraction
Errors in Radio Height Finding
8.4.1. Introduction
As a radio ray passes through the atmosphere, the length and direction
of its path varies with the radio refractive index. Uncorrected radar
output determines the position of a target by a straight line path at con-
stant velocity. The difference between the straight path and the actual
path results in an error which becomes increasingly significant as the
distance to the target increases. The height error (the component of the
position error normal to the surface of the earth) constitutes over 95 per-
cent of the total error. Until recently, the range of height finding equip-
ment was sufficiently limited so that the refraction errors could be either
neglected, or approximated by the effective (four-thirds) earth's radius
correction [27].
RAY THEORY 357
Bauer, Mason, and Wilson [28] obtained an equation for accurately
estimating radar target heights in a specific exponential atmosphere.
Beckmann [29] presented a i)robability estimate of the height errors with-
out meteorological measurements.
The purpose of the study is to investigate the correlation between
available meteorological ])arameters and height errors for targets of
interest in terminal air traffic control and to develop height error correc-
tion procedures using these parameters.
8.4.2. Refractive Index
The radio refractive index, n, of a propagation medium is the ratio of
the free space velocity of light, c, to the velocity in the medium, v, (i.e.,
n = c/v). Since the propagation velocity of the atmosphere is only
slightly less than the free space velocity, it is often convenient to use the
scaled up difference between the refractive index and unity. This ciuan-
tity is the refractivity.
The refractivity is obtained from (1.20). Normally, the equation for
N is dominated by the first term so that the refractivity can be approxi-
mated by an exponential function of height as shown in section 3.8.
8.4.3. Ray Theory
If the refractive index is assumed to satisfy (for a spherically strati-
fied atmosphere)
I > - 7- («-^«)
then, for frequencies greater than 100 kc/s, the path of a radio ray is
determined by Snell's law for polar coordinates (3.1) of chapter 3 (see
fig. 8.27) and the bending angle, r, is determined from (3.2). The dis-
tance, d, along the surface of the earth is obtained from (3.62).
The length of the path is called the geometric range and is obtained by
R = I CSC dr, (8.30)
and the apparent or radio range is found by
Re = I n CSC e dr = R -\- N X 10"^ esc d dr. (8.31)
358 REFRACTION AND REFRACTIVE INDEX MODELS
Figure 8.27. Geometry of radio ray refraction.
RAY THEORY
359
Fku're 8.28. Effective earth's radium; (leoineli
(/eontetnj.
360 REFRACTION AND REFRACTIVE INDEX MODELS
Because the difference between Re and the true slant range, Rq, is small
compared to the height error, the slant range and radio range are assumed
to be identical to the geometric range, R.
The apparent height of the target, in figure 8.28, is obtained by solving
(ro + haY = ro^ + R"" + 2r, R sin 6^ (8.32)
for ha. The following form is useful for numerical calculations:
K = ^(^ + 2rosin^o) _ ^g 33^
ro + Vro^ -\- R{R -\- ro sm do)
The height error for a target at height h is found by
€, = ha - h (8.34)
which will always be positive if n decreases with height.
If the refractive index is known as a function of height the foregoing
procedure is useful for determining the height error when the true height
and the arrival angle of the ray are hypothesized. Unfortunately, it is
not applicable for obtaining the height error from the apparent position
of the target.
8.4.4. Use of the Effective Earth's Radius
The inaccuracy of the "four-thirds earth" correction stems mainly
from the assumption that all radio rays have the same constant curvature.
The accuracy would be greatly enhanced if an "average" effective radius
could be determined for each ray path.
The following expression, with the effective earth's radius denoted by
Te in figure 8.28,
(r, + hy = r,2 -\- R^ -\- 2re R sin ^o, (8.35)
can be combined with (8.32) and (8.34) to obtain
« + ^ - f = (~^)(- - -). (8.36)
Ve 2re \ 2 /Vo Ve/
The difference between the curvature of the actual earth and the "average"
effective earth for the ray path represents the "average" curvature of the
ray. Thus, if the ray curvature can be determined as a function of the
USE OF EFFECTIVE EARTH'S RADIUS 361
target position and the refractive index structure, (8.36) will provide a
simple formula for approximating the height error,
since for target heights (h < 70,000 ft) at the maximum range (R c^ 150
mi) to be considered
I ^ - -^ I < 20 ft,
Ve 2re
or less than one percent of the total height error.
If the curvature of a ray, 1/p, at any point on the path is called K, then,
from (8.29) and (3.20)
_ Wo cos 00 dn
^ ~ 71^(1 + h/ro) dK ^^-^^^
From (8.32) ignoring terms of the order l/r^f, one obtains
so that (8.38) becomes
J. no {R" - hiy " V "^ Vo) dn_ (8.40)
~ n' R / , \ dh '
To/
1 +
ro
The refractive index usually decreases with height so that the quantity
1/2
(-6"
no ^ .
n2 1 + ^ — ^
To
varies only slightly with height, and the curvature at a point on the ray
path can be approximated by
i^r
K^('-^^)"'\'i\. (8.41)
362 REFRACTION AND REFRACTIVE INDEX MODELS
Therefore, (8.37) becomes
e.^ ^^-^Q (8-42)
where g represents the "average" gradient on the ray path (i.e., g = dn/dh
at an intermediate point on the path).
Since g depends upon the meteorological conditions along the path,
the basic problem is to determine g for a given target from the conditions
at and/or near the surface.
8.4.5. Meteorological Parameters
Measurement of the refractivity at the radar site will provide an esti-
mate of the gradient if a model of the refractive index structure is assumed.
In the exponential model, for example,
n{h) = 1 + A^, exp (-c/i) X 10-«
where N ^ is the surface refractivity and c is a constant, then
-77 = —cNs exp ( — c/i) X 10~^
For a target at a height, ht, the average gradient along the ray path is
g = -^[1 - exp i-cht)] X 10-', (8.43)
lit
but since ht is not known, g must be approximated as a function of the
apparent height.
Additional meteorological measurements at a sufficient height above
the surface to obtain values significantly different from the surface values
can be used to determine the initial gradient,
^-1 '- = !'" (If) xi«'' («■«'
where Nh is the refractivity at the height, H, of the above surface meas-
urements. The initial gradient provides a boundary condition for esti-
mating gf as a function of the apparent height. For convenience the initial
gradient of refractivity. Go = go X 10^ with H in kilometers, was used
as a prediction parameter.
ESTIMATION OF AVERAGE GRADIENT 363
For the purposes of this study, the average (per kilometer) gradient of
the first kilometer of the atmosphere is the only prediction parameter
used which will require upper air measurements. The average 1 km
gradient,
AN = Ni - Ns, (8.45)
where A'^i is refractivity at 1 km above the surface, was selected because
climatological summaries, as in chapter 4, can be used to estimate the
height error when meteorological measurements are unobtainable.
8.4.6. Calculation and Correlation of Height Errors
Bean, Gaboon, and Thayer [18] selected refractive index profiles, deter-
mined from radiosonde observations at 13 climatically distinct locations,
which represent a wide variety of mutually exclusive profile types. The
ray paths at arrival angles varying from 0° to near 90° were determined
for each profile by numerical evaluation of (8.29) through (8.33) using
methods similar to those described by Bean and Thayer [1]. The height
errors were calculated with (8.33) and (8.34) at selected height intervals
of 70,000 ft for each ray path. Newton's method of interpolation was
used to determine height errors for fixed ground distances to 150 mi. The
limits of height and distance were chosen to extend beyond the current
needs in terminal air traffic control, but are sufficiently restricted to allow
some of the previous assumptions.
The prediction parameters, N s, Go, and AA'^, were obtained from each
of the refractive index profiles. Linear and multiple regression analysis
were employed to obtain least squares estimates of the height error at
each height and distance for each prediction parameter and for various
combinations of the parameters.
8.4.7. Estimation of the Average Gradient
Based on the correlations the following forms, suggested by (8.43), were
selected for approximating g:
?■ = J^^fniha), (8.46)
92 = ^/2l(/la) -^-—-Mha) (8.47)
or
^3 = ^Mha) + ^Mha) + -J^Miha) (8.48)
depending upon the availability of Go and AA'^.
364
REFRACTION AND REFRACTIVE INDEX MODELS
To obtain a direct estimate of the height error, (8.46) through (8.48)
were combined with (8.42) and the functions /ij {i > j = 1, 2, 3) were
determined as least squares polynomials.
8.4.8. Regression Analysis
The volume of data processed is of sufficient magnitude that it is im-
practical to include it all in this report. Therefore, certain information
obtained from the regression analysis was selected as being the most
significant.
The mean height error provides the best general estimate obtainable
if meteorological data are not available. The standard deviation (about
the mean) of the height errors determines the reliability of this estimate,
since 68 percent of the observed height errors are within ±1 standard
deviation of the mean height error if the observations are normally dis-
tributed. In figure 8.29, the mean height error was plotted for each
target position, then contour lines were drawn to display the mean height
error as a function of true height and distance. By similar construction
the standard deviation of the height error as a function of target position
is displayed in figure 8.30.
The standard error of estimate establishes the same confidence limits
for prediction with a regression equation as the standard deviation does
for the mean. Thus, comparison of the standard error to the standard
deviation indicates the improvement in accuracy of prediction with
500 1750 2000
40 150 IGO
DISTANCE (MILES!
Figure 8.29. Mean height errors in feet.
REGRESSION ANALYSIS
365
2 5 10 20 30
250 300
00 no 120 liO 140 ISO 160
DISTANCE (MILES)
Figure 8.30. Standard deviation of height errors.
meteorological parameters. The standard error as a function of target
position for
eh = hiNs + a
is shown in figure 8.31; for
eh = biNs + hiGo + a
in figure 8.32; and for
eh = biNs + b2Go + 63 AA^ + a
(8.49)
(8.50)
(8.51)
in figure 8.33.
The regression equations, (8.49) through (8.51), were used because the
respective figures demonstrate how each additional parameter enhances
the accuracy of the estimate. The parameters Go and AA'^ applied indi-
vidually, that is,
and
€h = bOo + a
eh = bAN + a,
were of significant value only for targets at low heights (h < 10,000 ft).
Examination of the figures shows that prediction of eh with A'^., provides
significant improvement over the mean for target heights above 15,000 ft.
The addition of Go improves the estimate for heights below 15,000 ft
and the addition of AA'^ provides a slight overall improvement.
366
REFRACTION AND REFRACTIVE INDEX MODELS
In figures 8.29 through 8.33 the contours do not extend below 15,000 ft
for distances greater than 120 mi and 10,000 ft for distances greater than
80 mi. Correlations were not calculated for these target positions, be-
cause for certain refractive index profiles they are beyond the radio
horizon and for certain other profiles the arrival angle is too low for the
ray to penetrate a trapping layer. If conditions exist, without violating
the assumptions of sections 8.4.2. through 8.4.4. such that a target at
5000 ft height and 150 mi distance would be visible to radar, the resulting
height error would be about 10,000 ft.
Figure 8.31. Standard error of eh versus Ng
40 150 160
Figure 8.32. Standard error of th versus Ng and Go.
REGRESSION ANALYSIS
367
120 150 HO 150 160
Figure 8.33. Standard error of «h versus Ns, Go, and AN.
As an aid to further studies the coefficients for (8.49) through (8.51)
are Hsted in tables 8.9 to 8.17.
Table 8.8. The coefficients sm for {8.62) through {8.57)
-19.596
-17.849
-15.319
. 014096
.011202
. 006388
0. 77906 X10-<
. 13665X10-3
. 18549X10-3
0.67545X10-6
. 58925X10-2
.39074X10"'
-0.64975X10-2
-.55818X10-2
0. 12340X10-3
.12671X10-3
-0. 023980
-0.22547X10-*
Table 8.9
Constant term, a, in the regression equati
on {8.62)
Distance,
Height, kft
mi
5
10
15
20
25
30
35
40
50
60
70
5....
10....
15
20
25
30
35
-3
-14
-31
-55
-86
-123
-166
-215
-269
-327
-454
-586
-709
-3
-13
-29
-53
-83
-119
-162
-211
-267
-329
-470
-633
-812
-1000
-1186
-1352
-1473
-2
-10
-24
-44
-69
-100
-136
-178
-226
-280
-404
-552
-722
-913
-1122
-1342
-1565
-1774
-1946
-2048
-1
-8
-19
-35
-56
-81
-111
-146
-186
-231
-335
-461
-607
-776
-965
-1176
-1404
-1644
-1889
-2123
-6
-15
-28
-45
-66
-91
-120
-153
-190
-277
-381
-505
-648
-812
-997
-1203
-1429
-1672
-1928
-4
-12
-23
-37
-54
-74
-98
-125
-156
-228
-315
-418
-539
-678
-836
-1014
-1215
-1434
-1674
1
-3
-9
-18
-29
-44
-60
-80
-102
-127
-187
-259
-345
-446
-563
-697
-850
-1022
-1214
-1428
1
-2
-7
-14
-24
-35
-49
-65
-83
-104
-153
-212
-284
-367
-465
-578
-707
-852
-1019
-1203
2
-4
-9
-15
-23
-32
-43
-55
-70
-103
-144
-193
-252
-320
-399
-491
-597
-718
-855
2
1
-2
-5
-10
-15
-22
-30
-39
-49
-73
-102
-137
-179
-228
-286
-352
-429
-618
-620
2
1
-1
-3
-7
-11
-16
-22
-29
-36
-54
-76
-103
-134
-171
-214
-264
-322
-389
-465
40 .
45
50. .
60.--.
70
80. -
90....
100
110
120
130
140
150
368
REFRACTION AND REFRACTIVE INDEX MODELS
Table 8.10. Constant term, a, in the regression equation (8.53)
Height, kft
Distance,
mi
5
10
15
20
25
30
35
40
50
60
70
5
-3
-3
-2
-1
1
1
2
2
2
10
-12
-12
-10
-8
-6
-4
-3
-2
—
1
1
15
-28
-29
-24
-19
-15
-12
-9
-7
-4
-2
-1
20
-50
-51
-43
-35
-28
-22
-18
-14
-9
-5
-3
25
-77
-80
-67
-55
-45
-36
-29
-23
-15
-10
-7
30
-110
-115
-97
-80
-65
-53
-43
-35
-23
-15
-11
35
-148
-156
-133
-110
-90
-73
-60
-48
-32
-22
-16
40
-190
-204
-174
-144
-118
-97
-79
-64
-43
-30
-22
45
-235
-257
-221
-183
-151
-124
-101
-82
-55
-38
-28
50
-284
-316
-273
-227
-187
-154
-126
-103
-69
-48
-36
60
-383
-449
-394
-329
-273
-225
-185
-151
-102
-72
-54
70
-473
-600
-536
-452
-376
-311
-257
-210
-143
-101
-76
80 _._
-537
-762
-925
-700
-880
-595
-797
-497
-637
-413
-531
-341
-441
-281
-363
-191
-249
-136
-178
-102
90
-133
100
-1076
-1075
-940
-797
-667
-556
-460
-317
-226
-170
110
-1189
-1275
-1141
-976
-822
-687
-571
-395
-283
-212
120
-1236
-1468
-1355
-1174
-996
-837
-697
-486
-349
-262
130
-1637
-1576
-1390
-1189
-1005
-840
-590
-425
-319
140
-1748
-1794
-1619
-1400
-1192
-1003
-708
-512
-385
150
-1765
-1989
-1855
-1630
-1399
-1183
-843
-612
-460
Table 8.11. Constant term, a, in the regression equation {8.54)
Height, kft
Distance,
mi
5
10
15
20
25
30
35
40
50
60
70
5
-2
-1
-1
1
1
2
2
2
10
-2
-8
-8
-6
-5
-3
-2
-2
—
—
1
15
-4
-18
-18
-15
-12
-10
-8
-6
-3
-2
-1
20
-7
-33
-33
-28
-23
-19
-15
-12
-7
-5
-3
25
-9
-51
-51
-44
-37
-31
-25
-20
—13
-9
-6
30
-12
-73
-74
-64
-54
-45
-37
-30
-20
-13
-10
35
-13
-98
-100
-87
-74
-62
-51
-42
-28
-19
-14
40
-13
-125
-130
-114
-97
-81
-67
-55
-37
-26
-19
45
-10
-5
21
70
152
-156
-187
-254
-318
-369
-164
-201
-285
-378
-478
-144
-178
-256
-346
-448
-123
-152
-221
-302
-395
-104
-129
-187
-257
-338
-86
-107
-156
-216
-285
-71
-88
-129
-179
-237
-48
-60
-88
-123
-164
-33
-42
-62
-87
-117
-25
50
-31
60
-47
70
-66
80
-88
90.
-396
-383
-578
-669
-560
-677
-499
-615
-431
-536
-366
-457
-306
-384
-213
-269
-152
-194
-115
100
-146
110
-310
-741
-795
-739
-652
-560
-473
-334
-241
-182
120
-163
-776
-906
-867
-776
-673
-573
-408
-296
-224
130
-754
-998
-996
-909
-797
-682
-491
-359
-272
140
—649
-1055
-1117
-1044
-929
-804
-485
-430
-327
150
-436
-1057
-1217
-1179
-1068
-933
-690
-5)0
-389
REGRESSION ANALYSIS
369
Table 8.12. Coefficient of Ns, bi, in the regression equation {8.62)
Distance,
Height, kft
mi
5
10
15
20
25
30
35
40
50
60
70
5
10
15
20
25
30
35
40
45
50
60
70
80
90
0. 0318
.1153
.2545
.4495
.7005
1.0078
1.3714
1. 7917
2. 2688
2. 8030
4. 0426
5.5119
7. 2127
0. 0313
.1059
.2303
.4046
.6291
.9040
1. 2295
1. 6060
2, 0336
2. 5129
3. 6268
4. 9494
6. 4794
8. 2133
10. 1439
12. 2529
14. 5240
0. 0296
.0944
.2025
.3541
.5494
.7889
1. 0728
1. 4018
1. 7762
2. 1968
3. 1787
4.3513
5. 7236
7. 2966
9. 0736
11.0533
13. 2284
15. 5836
18. 0866
20. 7053
0. 0281
.0848
.1795
.3123
.4835
.6934
.9424
1.2311
1. 5599
1. 9295
2. 7938
3. 8300
5. 0453
6. 4464
8. 0412
9. 8363
11.8360
14. 0399
16. 4440
19. 0333
0. 0267
.0770
.1608
.2784
.4300
.6159
.8366
1. 0923
1. 3837
1.7113
2. 4780
3. 3981
4. 4788
5. 7273
7. 1529
8. 7649
10. 5679
12. 5772
14. 7931
17.2171
0. 0257
.0705
.1454
.2504
.3858
.5519
.7489
.9774
1. 2377
1. 5305
2. 2155
3. 0380
4. 0046
5. 1225
6. 4005
7. 8482
9. 4723
11. 2895
13. 3002
15. 5220
0. 0248
.0652
.1326
.2271
.3489
.4983
.6756
.8812
1.1154
1. 3788
1. 9952
2. 7355
3. 6053
4.6119
5. 7634
7.0684
8. 5370
10. 1777
12. 0051
14. 0251
0. 0243
.0608
.1218
.2074
.3177
.4531
. 6136
.7998
1.0119
1. 2503
1.8085
2. 4788
3. 2666
4. 1779
5. 2207
6. 4028
7. 7337
9. 2192
10. 8816
12. 7148
0. 0236
.0543
.1056
.1776
.2704
.3841
.5190
.6754
.8536
1. 0539
1. 5223
2. 0845
2. 7442
3. 5078
4. 3804
5. 3687
6. 4808
7. 7254
9. 1125
10. 6504
0. 0231
.0498
.0943
.1568
.2373
.3359
.4529
.5885
.7429
.9163
1.3216
1. 8072
2. 3769
3. 0342
3. 7846
4. 6329
5. 5863
6. 6512
7. 8354
9. 1488
0. 0225
.0462
.0858
.1413
.2127
. 3003
.4042
.5244
.6613
.8150
1. 1738
1. 6032
2. 1060
2. 6854
100
110
120 -
130
140
150
3. 3456
4. 0909
4. 9257
5. 8573
6.8904
8. 0339
Table 8.13. Coefficient of Ns, bi, iri the regression equation (8.53)
Distance,
Height, kft
mi
5
10
15
20
25
30
35
40
50
60
70
5
10
15
20
25
30
35
40
45
50
60
70
80
0. 0285
.1014
.2226
.3915
.6072
.8686
1. 1741
1. 5217
1. 9089
2. 3325
3.2717
4. 2984
5. 3577
0. 0303
.1015
. 2202
.3862
.5997
.8603
1. 1680
1. 5225
1. 9235
2. 3701
3. 3976
4. 5945
5. 9423
7.4110
8. 9558
10. 5049
11.9716
0. 0292
.0922
.1972
.3444
.5430
.7661
1.0411
1.3591
1.7204
2. 1254
3. 0671
4. 1841
5. 4790
6. 9435
8. 5676
10. 3305
12. 1954
14. 1047
15. 9660
17. 6672
0. 0279
.0835
.1763
.3065
.4742
.6797
.9234
1. 2056
1. 5268
1. 8873
2. 7287
3. 7339
4. 9073
6.2511
7. 7696
9. 4588
11.3115
13.3104
15.4244
17.6007
0. 0267
.0762
.1588
.2746
.4239
. 6069
.8240
1.0755
1.3619
1. 6836
2. 4355
3. 3361
4. 3907
5. 6047
6. 9854
8. 5356
10. 2579
12. 1560
14. 2189
16. 4316
0. 0257
.0701
. 1441
.2479
.3817
.5457
.7403
.9658
1. 2227
1.5114
2. 1864
2. 9957
3. 9449
5. 0400
6. 2887
7. 6975
9.2713
11.0200
12. 9405
15. 0401
0. 0249
.0649
.1317
.2253
.3459
.4939
.6694
.8729
1. 1046
1. 3651
1.9743
2. 7052
3. 5629
4. 5537
5. 6849
6. 9635
8. 3974
9. 9945
11.7614
13. 7056
0. 0244
.0607
.1212
.2061
.3155
.4497
.6089
.7935
1.0037
1. 2400
1. 7928
2. 4560
3. 2348
4. 1346
5. 1625
6. 3258
7. 6323
9. 0868
10. 7087
12. 4899
0. 0237
.0543
.1053
.1768
.2690
.3821
.5162
.6716
.8486
1.0475
1.5126
2. 0705
2. 7249
3.4816
4. 3456
5. 3231
6. 4215
7. 6489
9. 0139
10. 5245
0. 0232
.0498
.0942
.1563
.2365
.3346
.4511
.5860
.7396
.9121
1.3152
1. 7981
2. 3643
3. 0171
3. 7620
4. 6035
5. 5484
6. 6027
7. 7737
9. 0705
0. 0227
.0463
.0857
.1410
.2122
.2994
.4028
.5226
.6589
.8119
1. 1691
1. 5965
2 0968
90
100
110
120
130
140
150
2. 6730
3. 3294
4.0699
4. 8989
5. 8234
6. 8477
7. 9804
370
REFRACTION AND REFRACTIVE INDEX MODELS
Table 8.14. Coefficient of Ns, bi, in the regression equation {8.54)
Height, kft
Distance,
ml
5
10
15
20
25
30
35
40
50
60
70
5
0.0091
0. 0231
0. 0259
0. 0265
0. 0265
0. 0262
0. 0260
0. 0258
0. 0255
0. 0252
0. 0248
10
.0217
.0689
.0744
.0720
.0687
.0652
.0619
.0589
.0540
.0504
.0474
15
.0418
.1447
.1551
.1478
.1391
.1303
.1218
.1141
.1016
.0923
.0850
20
.0682
.2497
.2677
.2539
.2378
.2214
.2058
.1915
.1683
.1511
.1378
25
.0991
.3830
.4119
.3902
.3646
.3386
.3138
.2912
.2542
.2268
.2058
30
.1325
.5430
.5873
.5567
.5198
.4821
.4462
.4133
.3595
.3196
.2891
35
.1660
.7281
.7933
.7532
.7033
.6520
.6030
.5579
.4843
.4295
.3878
40
.1971
.9358
1. 0291
.9795
.9151
.8483
.7843
.7253
.6287
.5567
.5020
45
.2232
1. 1641
1. 2939
1.2354
1. 1553
1.0712
.9903
.9155
.7929
.7014
.6318
50
.2418
1.4090
1. 5863
1. 5206
1. 4239
1. 3209
1.2213
1. 1290
.9772
.8637
.7775
60
.2501
1. 9361
2. 2485
2. 1771
2.0459
1. 9009
1. 7588
1. 6263
1. 4070
1.2423
1. 1170
70
.2231
2. 4816
2. 9993
2. 9439
2.7804
2. 5892
2. 3986
2. 2192
1. 9202
1. 6944
1. 5223
80
.2005
3.0016
3. 8207
3.8126
3. 6248
3. 3863
3. 1423
2. 9101
2. 5191
2. 2224
1. 9953
90
3. 4452
4. 6780
4. 7689
4. 5741
4. 2912
3. 9913
3. 7012
3. 2075
2. 8288
2. 5384
100
3. 7639
5. 5316
5. 7994
5. 6251
5. 3042
4. 9474
4. 5952
3. 9877
3. 5170
3. 1545
110
3. 9201
6. 3291
6. 8710
6. 7588
6. 4182
6.0094
5. 5936
4. 8629
4.2900
3.8464
120...
3. 9312
7.0067
7. 9475
7. 9627
6. 6279
7. 1738
6. 6971
5. 8369
5. 1516
4. 6173
130
7. 4937
8. 9769
9. 2092
8. 9193
8. 4423
7. 9028
6. 9123
6. 1053
5. 4713
140
7. 7293
9. 8930
10. 4569
10. 2723
9. 7952
9. 2175
8. 0909
7. 1551
6.4118
150
7. 7099
10. 6160
11. 6542
11.6633
11.2325
10. 6218
9. 3770
8. 3049
7. 4432
Table 8.15. Coefficient of Go, bj, in the regression equation {8.53)
Height, kft
Distance,
mi
5
10
15
20
25
30
35
40
50
60
70
5
-0. 0065
-0.0019
-0. 0008
-0. 0003
-0.0000
0.0001
0.0002
0. 0002
0.0003
0.0003
0.0004
10...
-.0268
-.0084
-. 0043
-.0024
-.0015
-.0009
-. 0005
-.0003
-.0000
.0001
.0002
15
-.0617
-.0196
-. 0102
-.0060
-.0039
-.0026
-. 0018
-. 0012
-.0006
-.0003
-.0001
20
-.1123
-. 0356
-.0187
-.0112
-. 0073
-.0050
-. 0035
-.0026
-.0015
-.0008
-. 0005
25. .-
-. 1807
-. 0571
-.0299
-.0179
-.0118
-.0081
-. 0057
-.0043
-.0025
-. 0016
-.0011
30
-. 2695
-.0846
-.0441
-.0264
-.0175
-.0120
-.0086
-.0064
-. 0039
-. 0025
-.0018
35
-. 3820
-.1191
-.0615
-.0368
-.0244
-. 0168
-.0120
-.0091
-.0055
-.0036
-.0026
40
-. 5228
-. 1616
-. 0827
-. 0493
-.0326
-.0224
-.0161
-.0122
-. 0075
-.0048
-.0035
45
-. 6968
-. 2132
-. 1080
-.0642
-.0423
-. 0291
-.0209
-.0158
-.0097
-.0063
-.0047
50
-.9019
-. 2765
-. 1382
-. 0817
-. 0537
-. 0369
-.0265
-.0200
-.0123
-.0081
-.0059
60
-1.4924
-.4437
-. 2160
-. 1260
-. 0822
-.0564
-.0404
-.0305
-.0187
-. 0123
-.0090
70
-2. 3492
-. 6870
-. 3238
-. 1860
-. 1201
-. 0820
-. 0586
-. 0441
-.0270
-.0177
-. 0130
80
-3.5911
-1.0399
-.4737
-. 2672
-.1704
-.1156
-.0822
-.0616
-.0374
-.0245
-.0179
90
-1. 5533
-.6835
-.3782
-. 2375
-. 1596
-.1128
-.0839
-. 0507
-.0330
-.0239
100
-2. 3002
-. 9795
-. 5258
-.3243
-. 2163
-. 1520
-. 1125
-. 0674
-.0437
-.0314
110
-3. 3840
-1.3993
-. 7309
-. 4438
-. 2917
-. 2030
-. 1491
-.0884
-. 0568
-.0406
120
-4. 9414
-1.9997
-1.0155
-.6002
-. 3891
-.2703
-. 1962
-.1148
-. 0734
-. 0518
130
-2. 8631
-1.4124
-.8152
-. 5219
-.3547
-. 2562
-. 1482
-. 0939
-.0658
140
-4. 1054
-1.9740
-1.1116
-.6963
-.4717
-.3347
-. 1909
-.1194
-. 0827
150
-5. 8817
-2. 7735
-1.5206
-. 9329
-. 6185
-. 4353
-. 2437
-.1515
-. 1037
REGRESSION ANALYSIS
371
Table 8.16.
Coefficient of Go, b2,
in the
regression equation (8.54)
Distance,
Height, kft
mi
5
10
15
20
25
30
35
40
50
60
70
5
-0. 0032
-. 0133
-.0309
-. 0573
-. 0942
-. 1442
-.2105
-.2974
-.4101
-. 5553
-. 9784
-1.6559
-2.7138
-0. 0060
-.0029
-. 0067
-. 0124
-. 0202
-. 0306
-.0442
-.0618
-. 0840
-.1130
-. 1951
-.3276
-. 5396
-. 8786
-1.4169
-2. 2638
-3. 5736
-0. 0002
-.0013
-.0031
-. 0056
-.0091
-. 0136
-.0194
-.0265
-. 0354
-. 0464
-. 0767
-. 1222
-.1916
-. 2981
-. 4631
-.7186
-1.1171
-1.7384
-2. 7042
-4. 1878
-0. 0001
-.0005
-.0012
-. 0022
-.0036
-.0055
-.0079
-.0109
-.0146
-.0193
-.0322
-. 0516
-.0809
-. 1261
-. 1907
-.2906
-. 4432
-.6752
-1.0330
-1.5853
-0.0000
-. 0002
-.0005
-.0011
-. 0017
-. 0027
-.0038
-.0053
-.0072
-.0095
-.0159
-. 0256
-.0401
-. 0622
-.0929
-.1416
-. 2097
-.3139
-.4716
-. 7079
0.0000
-.0001
-. 0002
-.0004
-. 0008
-.0012
-.0017
-. 0024
-.0034
-.0045
-. 0078
-.0129
-.0206
-. 0323
-.0488
-.0741
-. 1095
-. 1646
-. 2424
-.3584
0.0000
-. 0000
-. 0001
-.0002
-. 0003
-.0005
-.0007
-.0010
-.0015
-.0020
-. 0037
-. 0065
-.0107
-.0171
-. 0266
-. 0406
-.0622
-.0906
-. 1372
-. 1978
0.0000
-.0000
-.0000
-.0001
-.0001
-. 0002
-.0004
-.0006
-.0008
-.0012
-. 0022
-.0038
-.0063
-.0102
-.0160
-.0246
-.0371
-.0548
-.0810
-.1175
-0. 0000
.0000
-.0000
-.0000
-.0000
-.0001
-.0001
-.0002
-.0002
-.0004
-.0007
-. 0014
-.0024
-.0040
-.0065
-.0101
-.0153
-. 0229
-. 0339
-.0485
-0. 0000
.0000
.0000
.0000
.0001
.0001
.0001
.0001
.0002
.0002
.0001
-.0000
-.0004
-.0010
-.0020
-. 0035
-.0059
-.0093
-.0142
-. 0212
0000
10
15
20
25
30
35
40
45
50
60
70
80
90
100
110
120
130
140
150
.0000
.0000
.0000
.0000
-0000
-.0000
-.0000
-.0001
-.0001
-.0002
-.0003
-.0006
-.0010
-.0017
-.0026
-.0039
-.0059
-.0085
-. 0123
Table 8.17. Coefficient of AN, hs, in the regression equation {8.54)
§1
Height, kft
'c« p
Q
5
10
15
20
25
30
35
40
50
60
70
5
-0. 0806
-0. 0301
-0. 0136
-0. 0060
-0.0011
0. 0021
0.0043
0.0057
0.0073
0.0083
0.0087
10
-. 3321
-. 1360
-.0739
-.0481
-.0313
-. 0201
-.0127
-.0075
-.0012
-.0024
-.0044
15
-. 7533
-.3146
-. 1754
-.1189
-.0819
-.0575
-.0411
-. 0296
-.0154
-.0077
-.0029
20
-1.3470
-. 5687
-.3196
-.2191
-. 1535
-.1104
-.0813
-.0608
-. 0355
-.0218
-.0132
25
-2. 1168
-.9026
-. 5084
-.3498
-. 2468
-. 1793
-. 1336
-.1015
-.0616
-.0401
-. 0265
30
-3. 0664
-1.3217
-. 7448
-. 5125
-. 3628
-. 2648
-. 1986
-.1519
-. 0940
-.0628
-.0430
35
-4. 1993
-1.8327
-1.0320
-. 7091
-. 5028
-.3679
-.2768
-. 2126
-. 1330
-.0901
-.0627
40
-5. 5178
-2.4440
-1.3744
-.9818
-.6680
-. 4894
-. 3690
-.2842
-. 1788
-. 1222
-.0860
46
-7.0221
-3. 1634
-1.7769
-1.2136
-.8604
-. 6308
-.4761
-.3672
-. 2320
-. 1592
-.1127
50
-8. 7091
-4. 0036
-2. 2458
-1.5276
-1.0819
-. 7935
-. 5992
-. 4625
-. 2929
-. 2016
-. 1434
60
-12.5871
-6. 0883
-3.4103
-2. 2978
-1.6230
-1. 1893
-.8979
-.6937
-. 4402
-.3038
-.2171
70
-16.9766
-8. 8019
-4. 9354
-3. 2912
-2.3149
-1.6931
-1.2773
-.9865
-.6263
-.4320
-.3092
80
-21.4838
-12.2504
-6. 9080
-4. 5605
-3. 1905
-2.3271
-1.7521
-1.3526
-. 8573
-. 5909
-.4227
90
-16.5206
-9. 4378
-6. 1745
-4. 2931
-3.1194
-2. 3426
-1.8054
-1.1417
-. 7843
-. 5608
100
-21. 6282
-12.6475
-8. 2077
-5.6667
-4. 1012
-3. 0720
-2. 3636
-1.4910
-1.0207
-.7284
110
-27.4311
-16.6690
-10.7799
-7. 4021
-5.3291
-3. 9747
-3. 0501
-1.9168
-1.3061
-.9311
120
-33. 4941
-21.6151
-14.0135
-9. 5614
-6. 8460
-5.0972
-3. 8960
-2. 4355
-1.6531
-1.1731
130
-27. 5400
-18.0521
-12.2761
-8. 7510
-6. 4662
-4. 9325
-3. 0686
-2.0720
-1.4666
140
-34.3121
-23. 0424
-15.6717
-11.1154
-8. 1908
-6.2123
-3. 8447
-2. 5769
-1.8159
150
-41. 4796
-29.0964
-19.9016
-14.0671
-10.3024
-7. 7822
-4. 7800
-3. 1896
-2. 2378
372 REFRACTION AND REFRACTIVE INDEX MODELS
8.4.9. Height Error Equations
The equations for approximating e^ were determined as
eh, = aio jrj + -B~ 9i + k, (8.52)
D D"^
eh, = a2o -7-7 + -jB" ^2 + k, (8.53)
and
en, = 030 7— H — ^ (?3 + A; (8.54)
where
D = R'' - 0.03587 /i^'.
Values for gi from (8.46) through (8.48) are found from
fiiiha) = a,i + a^-Jia + a,3/ia^ , («' = 1, 2, 3) , (8.55)
/i2(/ia) = an + a,5/ia , (z = 2, 3) , (8.56)
and
fzzQia) = age + a^^ha (8.57)
for R in miles and /la in thousands of feet. The term in D/h^ was intro-
duced to account for, in part, a large negative constant term which tended
to produce negative height errors for ranges less than 30 mi. Further-
more, the inclusion of this term increased the accuracy of the estimate of
thi by about 2 percent. An additional term in hj' for (8.55) increased
in the accuracy by about 1 percent but introduced a fictitious minimum
near 60,000 ft, while a term in ha~ for (8.56) and (8.57) increased the
accuracy of (8.53 and (8.54) by less than 0.1 percent. The relative im-
provement of (8.53) over (8.52) is about one percent. The coefficients
aij are listed in table 8.8. The constant term, k, which would vanish
if the equations were exact, is about —70 for a least squares approxima-
tion.
REFERENCES 373
8.4.10. Conclusions
Height-error correction can be significantly imi^roved by accounting
for the surface refractivity at the radar site. The use of the initial grad-
ient, in addition to the surface refractivity, yields a significant improve-
ment only for targers beyond about 60 mi and below 15,000 ft. In this
case, Go is important not only to improve the accuracy but to deter-
mine if the assumption in section 2.2 has been violated, namely, if
Go < — lOV^'o- The still further improvement obtained with the use of
AA'' would not, in general, justify the trouble and exi:)ense of measuring
this parameter.
If the distance to the target exceeds about 50 mi, the normal decrease
with height of the gradient should be accounted for in a height error
correction.
8.5. References
[1] Bean, B. R., and G. D. Thayer (May 1959), On models of the atmospheric refrac-
tive index, Proc. IRE 47, No. 5, 740-755.
[2] Misme, P. (Nov.-Dec. 1958), Essai de radiocHmatologie d'altitude dans le nord
de la France, Ann. Telecommun. 13, Nos. 11-12, 303-310.
[3] Tao, K., and K. Hirao (1960), Vertical distribution of radio refractive index in
the medium height of the atmosphere, J. Radio Res. Lab. 7, No. 30, 85-93.
[4] Misme, P. (Nov.-Dec. 1960), Quelques aspects de la radio-climatologie, Ann.
Telecommun. 15, 266.
[5] Bean, B. R., and G. D. Thayer (Aug. 1960), Rebuttal to P. Misme's comments on
"Models of the Atmospheric Radio Refractive Index," Proc. IRE 48, No. 8,
1499-1501.
[6] Craig, R. A., I. Katz, R. B. Montgomery, and P. J. Rubenstein (1951), Gradient
of refractive modulus in homogeneous air, potential modulus, Book, Propaga-
tion of Short Radio Waves, ed. D. E. Kerr, pp. 198-199 (McGraw-Hill Book
Co., Inc., New York, N.Y.).
[7] Zhevankin, S. A., and V. S. Troitskii (1959), Absorption of centimeter waves in
the atmosphere, Radioteknika i Electronika 4, No. 1, 21-27.
[8] Minzner, R. A., W. S. Ripley, and T. P. Condron (1958, U.S. extension to the
ICAO standard atmosphere, Tables and Data to 300 Standard geopotential
kilometers, U.S. Government Printing Office, Washington, D.C. 20402.
[9] Ratner, B. (1945), Upper air average values of temperature, pressure, and relative
humidity over the United States and Alaska (U.S. Weather Bureau).
[10] Bean, B. R., and J. D. Horn (Nov.-Dec. 1959), The radio refractive index near
the ground, J. Res. NBS 63D (Radio Prop.), No. 3, 259-273.
[11] Schulkin, M. (May 1952), Average radio-ray refraction in the lower atmosphere,
Proc. IRE 40, No. 5, 554-561.
[12] Bean, B. R., J. D. Horn, and A. M. Ozanich, Jr. (1900), CUmatic Charts and
Data of the Radio Refractive Index for the United States and the World,
NBS Mono. 22.
[13] W(mg, M. S. (Sept. 1958), Refraction anomalies in airborne propagation, Proc.
IRE 46, No. 9, 1628-1639.
[14] Report of Factual Data from the Canterbury Project (1951), Vols. I-III, (Dept.
of Scientific and Industrial Research, Wellington, New Zealand).
374 REFRACTION AND REFRACTIVE INDEX MODELS
[15] Freehafer, J. E. (1951), Tropospheric refraction, Book, Propagation of Short
Radio Waves, ed. D. E. Kerr, pp. 9-22 (McGraw-Hill Book Co., Inc., New
York, N.Y.).
[16] Bean, B. R. (July-Aug. 1959), Climatology of ground-based radio ducts, J. Res.
NBS 63D (Radio Prop.), No. 1, 29.
[17] Bean, B. R., and B. A. Cahoon (Nov. 1957), The use of surface weather observa-
tions to predict the total atmospheric bending of radio waves at small elevation
angles, Proc. IRE 45, No. 11, 1545-1546.
[18] Bean, B. R., B. A. Cahoon, and G. D. Thayer (1960), Tables for the statistical
prediction of radio ray bending and elevation angle errors using surface values
of the refractive index, NBS Tech. Note 44.
[19] Bean, B. R., and G. D. Thayer (May-June 1963), Comparison of observed
atmospheric radio refraction effects with values predicted through the use of
surface weather observations, J. Res. NBS 67D (Radio Prop.), No. 3, 273.
[20] Anway, A. C. (1961), Empirical determination of total atmospheric refraction at
centimeter wavelengths by radiometric means, Collins Res. Rept. CRR-2425
(Collins Radio Company, Cedar Rapids, Iowa).
[21] Anderson, W. L., N. J. Beyers, and R. J. Rainey (Aug. 1960), Comparison of
experimental and compeutd refraction, IRE Trans. Ant. Prop. AP-8, 456.
[22] Wilkerson, R. E. (July-Aug. 1962), Divergence of radio rays by the troposphere,
J. Res. NBS 66D (Radio Prop.), No. 4, 479.
[23] Norton, K. A., J. W. Herbstreit, H. B. Janes, K. O. Hornberg, C. F. Peterson,
A. F. Barghausen, W. E. Johnson, P. I. Wells, M. C. Thompson, Jr., M. J.
Vetter, and A. W. Kirkpatrick (1961), An experimental study of phase varia-
tions in line-of-sight microwave transmissions, NBS Mono. 33.
[24] Thompson, M. C, Jr. (1962), Private communication.
[25] Bennett, C. A., and N. L. Franklin (1954), Book, Statistical Analysis in Chemistry
and the Chemical Industry (John Wiley & Sons, Inc., New York, N.Y.).
[26] Brooks, C. E. P., and N. Carruthers (1953), Handbook of Statistical Methods in
Meteorology (Her Majesty's Stationery Office, London).
[27] Schelleng, J. C, C. R. Burrows, and E. B. Ferrell (Mar. 1933), Ultra-short-wave
propagation, Proc. IRE 21, No. 3, 427-463.
[28] Bauer, J. R., W. C. Mason, and F. A. Wilson (1958), Radio Refraction in a Cool
Exponential Atmosphere, Tech. Rept. 186, Lincoln Laboratory, Massachusetts
Institute of Technology, Cambridge, Mass.
[29] Beckmann, P. (1958), Height errors in radar measurements due to propagation
causes, Acta Technika, 3, No. 6, 471-488
Chapter 9. Radio-Meteorological Charts,
Graphs, Tables, and Sample Computations
9.1. Sample Computations of Atmospheric
Refraction
The following problem will serve to illustrate the a})plication of the
various methods of calculating bending of a radio ray as described in
chapter 3. A particular daily set of RAOB readings from Truk in the
Caroline Islands yields the following data:
Height above the
surface (km)
0.000
0.340
0.950
3.060
4.340
5.090
5.300
5.940
6.250
7.180
7.617
9.660
10.870
What is the total bending up to the 3.270-km level at initial elevation
angles of 0°, 10 mrad, 52.4 mrad (3°), and 261.8 mrad (15°) by (a) Schul-
kin's approach, (b) the exponential model, (c) the initial gradient method,
(d) the departures from normal method, (e) the use of regression lines,
and (f) the graphical method of Weisbrod and Anderson? Since the
gradient between the ground and the first layer is
AA^ 365.0 - 400.0 ,nooAr •. /i
71 = rv nAr. = — 102.9 A' umts/km,
An 0.340
and this is a decrease of N per kilometer that is less than the —157 N
units/km required for ducting, no surface duct is {)resent. However,
375
N value
(A units)
400.0 = N
365.0
333.5
237.0
196.5
173.0
172.0
155.0
152.0
134.0
125.5
98.0
85.0
376 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
should a surface duct have been present, it would have been necessary to
calculate the angle of penetration,
dp = V'2 [N, - Nh - 156.9 (Ah) (in km)],
to find the smallest initial elevation angle that yields a non-trapped ray.
Any initial elevation angle less than 9p cannot be used in bending calcula-
tions.
Schulkin's approach (a) of (3.13) yields the results shown in table 9.19
for mrad, table 9.20 for 10 mrad, table 9.21 for 52.4 mrad, and table
9.22 for 261.8 mrad, where dk+i is determined from (3.58) using
Tk = a -\- hk, and a is the radius of the earth,
a = 6370 km.
It should be remembered that dk = 0, 10, 52.4, or 261.8 mrad only for the
first-level calculation, and that thereafter 6k is equal to the dk+i computed
for the preceding layer; e.g., for the second layer of table 9.19
(^0 = Omrad), dk = 6.15 mrad,which is the ^t+icalculatedforthe first layer.
The exponential model solution (b) may be found by using tables
9.10 through 9.17. Interpolation will usually be necessary for A''^, do, and
height; this interpolation may be done linearly. In practice, one of these
three variables will often be close enough to a tabulated value that inter-
polation will not be necessary, thus reducing from 7 to 3 the number of
interpolations necessary. Since in the problem forA'^s = 404.9, h = 10.0 km
and 00 = 10 mrad
To. 10. 0(10 mrad) = 15.084 uirad
and at A = 20.00 km, do = 10 mrad.
To, 20.00(10 mrad) = 15.946 mrad,
and thus by linear interpolation for h = 10.870 km, do = 10 mrad,
TO, 10. 870, (10 mrad) = 15.084 + (15.946 - 15.084) 20 00 - 10 00
= 15.159 mrad.
COMPUTATIONS OF ATMOSPHERIC REFRACTION 377
Similarly for N s = 377.2 in the exponential tables,
To, 10. 870 (10 mrad) ~ 13.120 Hirad.
Again using linear interpolation, but now between the N s = 377.2 and
A^, = 344.5 atm, the desired value of r at 3.270 km for A^, = 360.0 and
^0 = 10 mrad is obtained.
Thus
400 — 377 2
T0.10.870(10 n.rad) = 13.120 + (15.159 - 13.120) 404 Q _ 377 2
= 14.798 mrad.
For the ^o = 0, 52.4, and 261.8 mrad cases, by similar calculations, using
linear interpolation :
7"o,io.87o, (0 mrad) = 21.386 mrad.
To, 10. 870, (52.4 mrad) = 5.816 mrad,
To, 10. 870, (261.8 mrad) = 1.270 mrad.
The initial gradient correction method (c) may be used if one deter-
mines the A^s* which corresponds to the observed initial gradient and then
applies (3.45). The initial A'' gradient is —102.9 N units/km, which, as
can be seen from table 3.17, corresponds to the N s = 450.0 exponential
atmosphere. Therefore, using the exponential tables of Bean and
Thayer [1]^ and (3.45) to determine the bending for the ^o = mrad case,
one finds by linear interpolation
Tio, 000(0) = Tio,ooo (400.0, mrad) + [rioo (450.0, mrad)
- Tioo (400.0, mrad)]
= 21.309 mrad + [5.908 - 3.657] mrad = 23.560 mrad.
The Tioo (400.0, mrad) is determined by linear interpolation between the
404.9 and 377.2 atm. At h = 20.0 km as given in the tables,
T2o,ooo (0 mrad) = T2o,ooo (400.0, mrad) + [tioo (450.0, mrad)
- r,oo (400.0, mrad)]
= 22.191 + [5.908 - 3.657] mrad
= 24.442 mrad.
^Figures in brackets indicate the literature references on p. 423.
378 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
Hence by linear interpolation,
.,...,. ,. „„. = 23.560 + [24.442 - 23.5601 ^oiooO I loioOO
= 23.637 mrad.
The bendings for ^o = 10, 52.4, and 261.8 mrad are as given below:
do = 10; no, 870 (10) = 15.053 mrad,
do = 52.4; 710,870 (52.4) = 5.864 mrad,
do = 261.8; Tio, 870 (261.8) = 1.280 mrad.
To use the departures-from-normal method (d) of determining bending,
it is first necessary to know the atmosphere which must be used for the
calculation. In the problem,
dN I
_ -^ = 102.9 A^ units/km,
Cl" I initial
which is within the range of the N ^ = 450.0 exponential atmosphere, as
can be seen from table 9.18. Thus one will use table 9.17 to determine
the d's and the r's in the N s = 450.0 exponential atmosphere, or one can
use the exponential atmosphere tables [1].
For an A^s = 450.0 atmosphere to a height of 10.870 km:
TNsio mrad) = 30.776 mrad,
TNsiio mrad) = 19.414 mrad,
rArs(52.4 mrad) = 7.024 mrad,
T7Vs(26i.8 mrad) = 1.506 mrad.
Equation (3.58) should be used for the 6 interpolation in preference to
linear interpolation, although, if no tables or other facilities are present
at the engineering site for easy acquisition of square roots, linear inter-
polation will suffice. Proceeding in table 9.17 with (3.58) for the first
layer at /i = 0.340 km and the ^o = mrad case:
V^2 ^ 2(n - ro) ^ ^q6 _ 2(^^ _ ^^)
ro
= 6.388 mrad. The remaining d's for the various layers are shown in
table 9.23. To determine the value of A at the bottom and top of the
COMPUTATIONS OF ATMOSPHERIC REFRACTION
379
layer, one makes use of (3.46) or figure 3.17. First, however, one must
determine the value of c in (3.46) to be used. Usually interpolation will
be necessary in table 9.18, but in the N s = 450.0 case it is not possible,
and thus the straight N s — 450.0 exponential atmosphere values are used.
From (3.46)
^(A^„ h) = N{h) + .¥, [1 - exp (-ch)],
and for the layer running from h = io h = 0.340 km, figure 3.17 yields
377.2 (1 - exp (cO)] = 0.0
450.0 [1 - exp (-C X 0.340)] = 32.8,
A (450.0, 0) = 400.0 + = 400.0
A (450.0., 0.340) = 365.0 + 32.8 = 397.8,
whence
AA = A (450.0, 0.340) - A (450.0, 0) = 397.8 - 400.0 = -2.2 A^ units.
Therefore, the departure term of (3.23),
and
and, therefore,
and
+
7/c+l L
AAiNs)
becomes
+ 6.388 L
2.2
= +0.689 mrad.
The remaining calculations are tabulated in table 9.23 for the ^o = mrad
case, in table 9.24 for the ^o = 10 mrad case, in table 9.25 for the ^o = 52.4
mrad case, and in table 9.26 for the ^o = 261.8 mrad case. The sum of
the departures for the mrad case is
+
/fc+i L
iN
AA (N.)
k+i
N,
= —5.335 mrad.
Determination of the bending is required in part (e) of the problem by
using regression lines. By (3.10), using table 9.7 and 9.8, it is found for
the 00 = mrad case that at 10.0 km (from table 9.7)
TO, ,0.0 = (0.1149) (400.0) - 18.5627 ± 7.5227
= 27.3973 ± 7.5227 mrad
380 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
and at 20.0 km (from table 9.8)
To.20.0 = (0.1165) (400.0) - 17.9573 ± 7.5131
= 28.6427 ± 7.5131 mrad.
Thus, by linear interpolation
Ti.2 = To.10.87 = 27.3973 + (28.6427 - 27.3973) 9q qq I |q qq ± 7.5227
^ .„ _<^„„ „_.„.. 10.87 - 10.00 ,
T (7.5227 - 7.5131) 20.00 - 10.00
Ti,2 = 27.5056 ± 7.5218 mrad.
Similarly, for the remaining 0's,
ri,2(io mrad) = 13.9548 ± 0.9701 mrad,
Ti,2(53.4 mrad) = 5.2186 ± 0.0817 mrad,
Ti, 2(261. 8 mrad) = 1.2695 ± 0.0158 mrad.
Determination of the bending by means of the graphical method (f)
of Weisbrod and Anderson yields, from figure 3.18 for 500 tan 6, for the
first layer.
At do At ^0 At ^0 At do
h(m.) = mrad = 10 mrad = 52.4 mrad = 261.8 mrad
0.000 0.0 5.0 26.2 134.0
0.340 3.0 5.8 27.0 134.0
which yields for the bending in the first layer,
At do At ^0 At ^0 At do
= mrad = 10 mrad = 52.4 mrad = 261.8 mrad.
11.67 3.24 0.66 0.13
Similarly, the bending for the entire profile may be obtained, and shown
to be
At do At do At ^0 At do
= mrad =10 mrad =52.4 mrad = 261.8 mrad.
24.42 14.00 5^32 iTlT
COMPUTATIONS OF ATMOSPHERIC REFRACTION
381
The answers to the several parts of the problem are summarized in the
next table of this chapter (immediately preceding the main body of
tables). Bending values for the assumed profile, from a method which
exponentially interpolated layers between given layers and then integrated
between resulting layer^, assuming only a linear decrease of refractivity
between interpolated layers, are included for the sake of comparison.
The comi^utations were performed on a digital computer.
The reason that the answers to i)art (e) vary so radically from the
remaining answers for the ^o == mrad case and not so much for the
^0 = 261.8 mrad case is the fact that the accuracy of the regression line
method increases with increasing initial elevation angle, ^o- It must be
remembered that the statistical regression technique, like the exponential
model, is an adequate solution to the bending problem for all ^o's larger
than about 10 mrad, and all heights above 1 km.
The reason that the answers in part (f) and part (a) agree more closely
than with any other of the answers is because (3.49) is, as mentioned
before, Schulkin's result with only the approximation, tan dk = dk for
small angles, omitted. For this individual profile, the bending obtained
from an exponential atmosphere does not give particularly accurate bend-
ings; however, for 22 five-year mean refractivity profiles, figure 3.9 shows
that exponential bending predicts accurately within 1 percent of the
average bending for these five-year means. Figure 3.19 shows the rms
error in predicting bending at various heights as a per cent of mean bend-
ing (not including superrefraction).
In summary, it is recommended that the communications engineer
either use the statistical regression technique or the exponential tables
[1] without interpolation (i.e., pick the values of height, N s, and Oq that
are closest to the given parameters) for a quick and facile bending result,
keeping in mind the restrictions on these methods. However, as men-
tioned before, use of Schulkin's method is recommended if accuracy is
the primary incentive.
Summary of refraction results for the sample computation
Prob-
lem
part
Method used
Bending
in mrad at
6o==0 mrad
Bending
in mrad at
So = 10 mrad
Bending
in mrad at
00=52.4 mrad
Bending
in mrad at
00=261.8 mrad
a.
Schulkin's method
24. 248
14.008
5.341
1.196
b.
Exponential model
21.386
14. 798
5.816
1 270
c.
Initial gradient correction method.
23. 637
15. 053
5.864
1.280
d.
Departures from normal method..
25. 441
14. 858
5.350
1.143
e.
Statistical regression method
27. 506
d=7. 522
13. 955
±0. 9701
5. 2186
±0. 0817
1. 2695
±0. 0158
f.
Graphical method
24.42
24 171
14.00
14. 104
5.32
5.343
1 168
Compa
inter
rison (exponential layer
polation) bending...
1. 168
382
CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
Tables 9.1 through 9.9 are tables of coefficients, a and h, standard errors of
estimate, SE, and correlation coefficients, r, for use in the statistical method
of refraction prediction.
Table 9.1.
Variables in the statistical method of refraction prediction
for h — hs = 0.1 km
So
r
b
a
SE
0.0
1.0
2.0
0. 2665
.2785
.2881
.3048
. 1915
.2070
.2100
.2105
.2105
.2107
.2108
0. 0479
.0257
.0162
.0073
.0053
.0025
.0009
.0005
.0002
.0001
. 00004
-8.7011
-4. 1217
-2. 2732
-.9181
-. 6085
-.2639
-.0973
-. 0507
-. 0250
-.0120
-.0040
6. 7277
3. 4363
2. 0960
5.0
0. 8792
)0.0
20.0
52.4
1.0551
0. 4555
. 1688 ,
100.
. 0879
200.0
400.0
900.0
.0435
.0208
.0070
Table 9.2.
Variables in the statistical method of refraction prediction
for h - ha = 0.2 km
00
r
b
a
SE
0.0
0. 2849
.2979
.3104
.3415
.2306
.2550
.2604
.2610
.2613
.2613
.2604
0. 05801
.0348
.0239
.0117
.0073
.0035
.0013
.0007
.0003
.0002
.00005
-10.4261
-5. 6431
-3. 5707
-1.5287
-0. 7436
-. 3184
-. 1162
-.0603
-. 0299
-. 0143
-.0047
7. 5726
1.0-
2.0.
4. 3330
2. 8357
5.0 .-
1.2512
10.0
1. 1990
20.0
52.4.
0. 5122
.1890
100.0
.0983
200.0
.0486
400.0
.0233
900. 0...
.0078
Table 9.3. Variables in the statistical method of refraction prediction
for h — ha = 0.5 km
flo
r
b
n
SE
0.0
0.3615
.3997
.4369
.5205
.3933
.4563
.4731
.4753
.4760
.4761
.4764
0. 0769
.0510
.0384
.0228
.0140
.0071
.0027
.0014
.0007
.0003
.0001
-14.6443
-9. 0567
-6. 5408
-3. 6605
-1.9055
-0. 8926
-.3308
-. 1721
-. 0851
-. 0408
-.0137
7. 6170
1.0
2.0
4. 4954
3. 0395
5.0
1. 4376
10.0
1. 2733
20.0
0. 5365
52.4
.1966
100.0
.1022
200.0
400. .
.0505
.0242
900.0
.0081
COMPUTATIONS OF ATMOSPHERIC REFRACTION
383
Table <).4.
Variables in the statistical method of refraction prediction
for h — he = 1.0 km
Bo
r
b
a
SE
0.0...
0. 3936
.4620
.5238
.6348
.5718
.6598
.6823
.6851
.6859
.6860
.6864
0. 0840
.0607
. 04918
.0337
.0224
.0124
.0049
.0026
.0013
.0006
.0002
-15. 1802
-10.3739
-8. 2066
-5. 4816
-3. 2378
-1. 6959
-0. 6495
-.3388
-. 1676
-. 0803
-. 0270
7 6151
1.0
2.0
4.5217
3 1040
5.0
1 5931
10.0
1 2574
20.0
5531
52.4
2071
100.0 . .
1080
200.0
0534
400. -
0256
900.0
0086
Table 9.5. Variables in the statistical method of refraction prediction
for h - he = 2.0 km
Bo
r
6
a
SE
0.0
0. 4524
0. 0985
.0752
.0636
.0475
.0345
.02111
.0089
.0047
.0023
.0011
.0004
-17.7584
-12.9451
-10. 7566
-7. 8969
-5. 3712
-3. 1571
-1.2770
-0. 6705
-0. 3323
-0. 1593
- .0535
7 5391
1.0. -
5490
6316
7707
7634
8515
8668
8679
8681
8682
8684
4 4420
2.0.
3 0277
5.0
1 5234
10.0.
1 1421
20.0
5086
52.4..
2003
100.0
1057
200.0...
0524
400. 0. .
0252
900.0
0084
Table 9.6.
Variables in the statistical method of refraction prediction
for h — hs = 5.0 km
Bo
r
6
a
SE
0.0
0. 4962
.6101
.7030
.8504
.8674
.9484
.9674
.9695
.9701
.9702
.9703
0.1115
.0881
.0764
.0601
.0464
.0308
.0139
.0075
.0037
.0018
.0006
-19. 1704
-14.3543
-12.1589
-9. 2514
-6. 6445
-4. 0706
-1.6236
-0. 8348
-.4098
-. 1960
-.0658
7 5676
1.0
2.0..
4. 4401
3 0001
5.0
1 4422
10.0
1 0420
20.0
4028
52.4..
1426
100.0
0739
200.0
0365
400.0
0175
900.0-
0059
384
CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
Table 9.7.
Variables in the statistical method of refraction prediction
for h - hs = 10.0 km
So
r
b
a
SE
0.0
0. 5099
.6290
.7250
.8734
.8950
.9723
.9907
.9927
.9931
.9932
.9932
0. 1149
.0915
.0799
.0635
.0498
.0338
.0157
.0085
.0043
.0020
.0007
-18.5627
-13.7469
-11.5514
-8. 6434
-6.0729
-3. 5012
-1. 1441
-0. 5084
-.2310
-. 1078
-.0359
7. 5227
1.0
2.0
4. 3895
2 9443
5.0
1.3733
10.0
0.9713
20.0
3179
52.4
.0844
100.0
.0406
200.0
.0197
400.0
.0094
900.0
.0032
Table 9.8.
Variables in the statistical method of refraction prediction
for h - ha = 30.0 km
e.
r
b
a
SE
0.0
0. 5155
.6367
.7336
.8815
.9028
.9785
.9968
.9984
.9986
.9986
.9986
0.1165
.0931
.0814
.0651
.0514
.0353
.0169
.0093
.0047
.0023
.0008
-17.9573
-13.1413
-10.9463
-8. 0397
-5. 4747
-2. 9228
-0. 6738
-. 1802
-.0467
-.0161
-.0048
7. 5131
1.0
2.0
4. 3763
2. 9281
5.0
1.3521
10.0
0. 9573
20.0
.2909
52.4
.0535
100.0
.0203
200.0
0096
400. 0- -
.0046
900.0
.0016
Table 9.9'.
Variables in the statistical method of refraction prediction
forh - h, = 70.0 km
e.
r
b
SE
0.0
0. 5174
.6391
.7361
.8837
.9051
.9797
.9979
.9997
1.0000
1.0000
1.0000
0. 1170
.0936
.0820
.0656
.0519
.0358
.0173
.0096
.0048
.0024
.0008
-17.9071
-13.0912
-10.8960
-7.9895
-5. 4209
-2.8696
-0. 6246
-. 1402
-.0212
-.0027
-.0002
7.5113
1.0
4. 3738
2.O.. . -
2. 9251
5.0
1.3481
10.0. -.
0. 9539
20.0
.2862
52.4
.0445
100.0
200.0
.0095
.0013
400.0
.0002
900.0
.0001
COMPUTATIONS OF ATMOSPHERIC REFRACTION
385
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COMPUTATIONS OF ATMOSPHERIC REFRACTION
387
3
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COMPUTATIONS OF ATMOSPHERIC REFRACTION
389
Table 9.18. Initial N gradients, ANe, in the CRPL
Exponential Reference Atmosphere*
Range of ANe
(N^ units/km)
N{h)
ANe <, 27. 55
27. 55 < ANe <, 35. 33
35. 33 < ANe ^ 47. 13
42. 13 < ANe <, 49. 52
49. 52 < AA^€ <, 59. 68
59. 68 < ANe <. 71. 10
71. 10 < ANe ^ 88. 65
88. 65 < ANe
200 exp(-0. 1184/1)
252.9exp(-0.1262h)
289 exp(-0. 1257/1)
313 exp(-0. 1428/1)
344. 5 exp(-0. 1568/i)
377. 2 exp( -0.1732/1)
404. 9 exp(- 0.1898/1)
450 exp(-0. 2232/1)
*Note height, h, is in kilometers.
390
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cococococococo cocococococo
cDcDcDCOCOcDCO cOCOcDcDcDcD
O O to O to O O O O O to o o
8'tofor^cDcocs toc^'^ioodto
cDcocoosr^t->. tot
-^ CO CO IM 1— " ^ t-l ^r
to to CO C^ OS 00
oooooQ ooor^oo
•^ to CO Tf OS O Tf to 00 >— I CD t^
COOSOCOOCO OS C^ 1-H CD CO 00
00 'coTjJioto toco'r^t^oscD
o^
SCO b- 00 OS «-< i-H i-H
COMPUTATIONS OF ATMOSPHERIC REFRACTION
391
<r; »c r^ cc CO O cs o c^ --" fo — <
»0 C^ 00 Oi ^ O <M CM CO C^ *0 O
r-- uo '^ QC O CM CO Tji CD 00 CN CO
; CO O '^f i^D O *0
(N CO ilO Oi ^ <N CC TTiOt^Oii:
O ''ti ^ b» W3 lO 00 iCTJOOC^lOO
* ■ -- C3 CC (M M PO
00 CO t^ Oi CO
(N CC 00 O CN C^l Tf
"^ CD t-* •-* Tf
OiOOcDcDiDCD cCcDXr-t^
.— I M O Oi 1— ' OS CD O <—< CD ■^ r^
oO'^O'^cso cooor-o
OO^QOOOOOCS ^CDOOr^CM
r-^oicct--ooas^ c^Tj^io^io
<N (N CC CC CO CC -* TP Tf Tfi 40 iO
lo ^ 00 oi c^i 00 r^
t^ lO CO CO CO 00 t^
CD .— I <N <-• iC I
O Oi CD O CO CO O 05 O: CO
CM r^ O CD CO
tC-; r^ o C3
~ oi CO ■^ r^
CM ^ CD' CO
O CO CO •-H !--. C^ Tj< CD CD t^ lO CO
OOOOOOO O O -* CD O
00 CM CM CD O CM 00 CM CD t^ 00 C^
CD CM CM O to -^ CM CD 00 00 O "^
ooo ooo
"»r o CD ■^ OJ o
CO 05 O CO O CO
OOOC0-^"'0»0
_, t^ t-- t^ t^ r^ t^ r^
y CO CO CO CO CO
o o o r- o o
■^ >o 00 '-' CO r^
OS CM 1—1 CO CD 00
■^ OJ o ■^ »o 00 >— ' CO r^
r^ t-, t^ r^ r^ 00
-_-.-,-- cocococococo
cDcDcDcDcDcDcD cDcOcDCOCOcD
O O »0 O *C O O 0*000
OOCOt-^CD'cOCM tOCMTTiOodiO
O CD CO CO OS r^ t^ »0 iO CO CM 05 00
'^ CO CO CM ■-H '-' ^ ^^^^
OOQOQO OOOt-OO
rrtOcp-^cTiO ■^»O00'— 'COt-^
O CO Ol O CO O CO 05 CM »-H CO CD 00
'co-'TuO'O tocD'r^t^oio
Oi-H CM
O'—CMCO'^u^cO t>.00Oi-— "■—".-«
^
■^ O X ■^ Oi -^ "^ -— ' 00 CM OO Oi
COC^COiOOOOCD .— icocoo-^
^^co^OOO 000.-H0
•OCMCM-^CDCD-^ 00CMO"^CD
COOiCD*- "01C0*O doOOO'^CM
0001»OCOt^OCM TTt^i— 'OOr-
'-H'—'CMCOCO'^''^ TT-^iOiCCO
cDcDCDcDcDCOcO cDcCOCOCD
CM CM CM CM CM
CM CM CM CM CM
O lO Ol 00 lO CD CM ■* t- CO to r-
t>.,— iQ— 't^os-H aocooscsic
00 1-1 O CD Oi O Tf »0 O CM CO O
^CMCOCOCO-^TfJ •^t'lOiOco't^
COcDcOCOcDcCcD CDcDcDcDcO
CM CM CM CM CM CM CM CM CM CM CM CM
to ^ CO ^ CM O
t^ O t-^ CTl 00 -^ C
*o t-- 1— I Tj< CD r- '—
00 00 Oi 05 OS Ol OS
■^ O O CO OS
O CD 00 CD 1-H
O CM CO OS CO
(_ ocDcococDcDos i>-r^r-t^i>.
iO.-<00OSCM00h^ ■^<— iiOOOCO
t^ »o CO ^ CO 00 r^ CMr^ococo
CD i-H CM —< lO to C
O O) CD O CO CD <:
^ -H CO ■* CM C
r^ ^ t^ OO)
OS OS CO -^ t^
O CO CO ^ t^ CM ■■
r- CD OS 00 TtH :
CD' CO I^ to CO
OOOOOOO O O ^ CO o
00 CM CM CD O CM 00 CM CD b* 00 CM
CO 00 00 O Tji
— I -^ CM
OOOOOO
■"ff to CO ^ OS o
CO OS O CO O CO
_____.^ OOOb-OO
■■(titOCO^OSO ■^tOOO'-HCOt^
— — * OSCM.-HCDCO00
O o o CO -^ to lO to :o r^ r^ OS o
CO CO CO CO CO CO CO CO CO CO CO CO CO
COcDCDCDCOCOcD cDcOCOcOCOCO
o o to o to O O O O O to o o
CJtocor-^CDCOCM tOCM-^J'toodtO
O CO CO CO OS t^ I-- to to CO CM OS 00
-* CO CO CM ^ ^ ^ ^^,—1^
O O O O O Q O o o t^ o o
■^tocD'^QsO ■^tOOO'— "COt^
COOSOCOOCO OSCM-— 'CDCDOO
3 ' ' CO "^ to to to CO r^ r^ OS o
O ^ CM
t^ 00 OS .-H r-H r-^
392
CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
Table 9.23. Departures method sample computation for 60 = 10 mrad
h
N(h)
A(N,h)
A.4
e
2
De-
parture
term
TNsih)
'"1.2
400.0
365.0
333.5
237.0
196.5
173.0
172.0
155.0
152.0
134.0
125.5
98.0
85.0
400.0
397.8
419.5
459.7
475.7
478.5
484.1
485.6
490.5
493.4
493.3
495.9
495.2
-2.2
+21.7
+40.0
+ 16.0
+2.8
+5.6
+1.5
+4.9
+2.9
-.1
+2.6
-.7
6.388
11.230
22. 683
28. 195
31. 383
32. 191
34. 654
35.811
39. 123
40. 581
47. 236
50. 645
0.3131
.1135
.0590
.0393
.0336
.0315
.0299
.0284
.0267
.0251
.0228
.0204
+0. 689
-2. 463
-2. 359
-0. 629
-.094
-.176
-.045
-.139
-.077
+.003
-.059
+.014
S = -5. 335
0.340
0.950
3.060
4.340
5.090
5.300
5.940
6. 250
7.180 .
7.617
9.660
10. 870
30. 776
25. 441
Table 9.24. Departures method sample computation for do = 10 mrad
h
N(h)
A(N,h)
^A
e
2
De-
parture
term
TN>(h)
Ok+Bk+i
400.0
365.0
333.5
237.0
196.5
173.0
172.0
155.0
152.0
134.0
125.5
98.0
85.0
400.0
397.8
419.5
459.7
475.7
478.5
484.1
485.6
490.5
493.4
493.3
495.9
495.2
-2.2
+21.7
+40.0
+16.0
+2.8
+5.6
+1.5
+4.9
+2.9
-.1
+2.6
-.7
10.000
11.873
15. 038
24. 789
29. 947
32. 938
33. 712
36. 069
37. 181
40. 383
41.801
48. 282
51. 630
0. 0914
.0743
.0502
.0365
.0318
.0300
.0287
.0273
.0258
.0243
.0222
.0200
+0. 201
-1.613
-2. 008
-.585
-.089
-.168
-.043
-.134
-.075
+.002
-.058
+.014
2 = -4. 556
0.340
0.950
3. 060
4.340 . -
5.090
5 300
5 940
6.250
7.180
7.617
9 660
10.870
19.414
14. 858
Table 9.25. Departures method sample computation for do = 52.4 mrad (3°)
h
N{h)
A(N,,h)
A.4
e
2
De-
parture
term
w,(/l)
Bk+Bk+i
400.0
365.0
333.5
237.0
196.5
173.0
172.0
155.0
152.0
134.0
125.5
98.0
85.0
400.0
397.8
419.5
459.7
475.7
478.5
484.1
485.6
490.5
493.4
493.3
495.9
495.2
-2.2
+21.7
+40.0
+ 16.0
+2.8
+5.6
+ 1.5
+4.9
+2.9
-.1
+2.6
52.4
52. 750
53. 550
57. 061
59. 567
61.045
61.477
62. 792
63. 433
65. 368
66. 277
70. 512
72.920
0. 0190
.0188
.0181
.0171
.0166
.0163
.0161
.0158
.0155
.0152
.0146
.0139
+0. 042
-.408
-.724
-.274
-.046
-.091
-.024
-.078
-.045
+.002
-.038
+.010
2 == -1.674
0.340
0. 950
3 060
4.340
5.090
5.300
5. 940
6 250
7 180
7.617
9 660
10.870.
7.024
5.350
EXPONENTIAL REFERENCE ATMOSPHERE 393
Table 9.26. Departures method sample computation for do = 261.8 mrad {15°)
h
mh)
AiNs,h)
AA
e
2
De-
parture
term
TN.ih)
dk+eic+i
400.0
365.0
333.5
237.0
196.5
173.0
172.0
155.0
152.0
134.0
125.5
98.0
85.0
400.0
397.8
419.5
459.7
475.7
478.5
484.1
485.6
490.5
493.4
493.3
495.9
495.2
-2.2
+21.7
+40.0
+16.0
+2.8
+5.6
+1.5
+4.9
+2.9
-.1
+2.6
-.7
261.8
261. 880
262. 035
262. 758
263. 308
263. 633
263. 733
264. 034
264. 184
264.647
264. 872
265. 934
266. 592
0.0038
.0038
.0038
.0038
.0038
.0038
.0038
.0038
.0038
.0038
.0038
.0038
+0.008
-.083
-.152
-.061
-.011
-.021
-.006
-.019
-.011
+.000
-.010
+.003
S = -0. 363
0.340—
0.950
3.060
4.340
5.090
5.300
5.940
6.250
7.180
7. 617_
9.660
10.870
1.506
1 143
9.2. Tables of Refraction Variables for the
Exponential Reference Atmosphere
The following table of estimated maximum errors should serve as a
guide to the accuracy of the tables.
Errors in elevation angle, 6:
do < 4: mrad ± 0.00005 mrad ) nearly
4 mrad < ^o < 100 mrad ± 0.000005 mrad > independent
do < 100 mrad ± 0.00004 mrad ) of A^,.
Errors in r, e (in milHradians) :
N, = 450 404.8 377.2 344.5 313 252.9 200
00 = ± 0.001 0.00065 0.0005 0.0004 0.0003 0.0002 0.0002
do = 1° 0.0003 0.00015 0.0001 0.00008 0.00006 0.00005 0.00005
00 = 3° 0.00004 0.000025 0.00002 0.000017 0.000015 0.000013 0.000012
Errors in Ro, R, Re, or Ah (in meters):
Ns = 450 404.8 377.2 344.5 313 252.9 200
?o = ±5.0 2.7 1.8 1.2 0.8 0.65 0.6
?o = 1° 0.4 0.3 0.25 0.2 0.17 0.15 0.14
Assume that the error in AR or ARe is ±0.5 percent or ±0.1 m, whichever
is larger.
394
CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
In tables 9.27 to 9.29 the following equations were used in determining
the various profile variables:
AA^ = 7.32 exp (0.005577 A^.),
Ce = hi
N..
Ns + AN
-dN, = CeNs,
k =
1 - -^ (ceNs) X 10-'
Us
ris = 1 + A^, X 10-«,
where ro is taken as 6373.024987 km for all values of N s in these tables.
Table 9.29 was prepared using an iterative method for solution of
transcendental equations involved. In all these tables the accuracy may
be tak(m as ±1 in the last digit listed.
Table 9.27. Parameters for exponential refraction with height for various values of Na
A^,
-AN
Ce
-dA^o
k
200
22. 3317700
23. 6125966
24. 9668845
26. 3988468
27. 9129385
29. 5138701
31. 2066224
32. 9964614
34. 8889558
36. 8899932
39. 0057990
41. 2429556
43. 6084233
46.1095611
48. 7541501
51. 5504184
54. 5070651
57. 6332884
60. 9388149
64. 4339281
68. 1295015
72. 0370324
76. 1686780
80. 5372922
85. 1564647
90. 0405683
0.118399435
. 119280212
. 120458179
. 121916361
. 123642065
. 125626129
. 127862319
. 130346887
. 133078254
. 136056720
. 139284287
. 142764507
. 146502381
. 150504269
. 154777865
. 159332141
. 164177379
. 169325150
. 174788368
. 180581312
. 186719722
. 193220834
.200103517
. 207388355
. 215097782
. 223256247
23. 6798870
25. 0488444
26. 5007993
28. 0407631
29. 6740955
31.4065323
33. 2442030
35. 1936594
37.2619112
39. 4564487
41. 7852861
44. 2569972
46. 8807620
49. 6664087
52. 6244741
55. 7662495
59. 1038565
62. 6503054
66. 4195799
70.4267116
74. 6878887
79. 2205420
84. 0434770
89. 1769927
94. 6430240
100.4653113
1. 17769275
210
1. 18991401
220
230
1.20315637
1. 21752719
240
250..
1. 23314913
1. 25016295
260
1.26873080
270
1. 28904048
280--
1.31131073
290
1. 33579768
300- -
1. 36280330
310
1.39268608
320
330
340
350
360
1.42587494
1. 46288731
1. 50435338
1. 55104840
1. 60393724
370
1. 66423593
380
390
1. 73349938
1. 81374807
400
410
420
430
440
450
1. 90765687
2. 01884302
2. 15232187
2.31525447
2. 51823286
2. 77761532
EXPONENTIAL REFERENCE ATMOSPHERE 395
Table 9.28. Parameters for exponential refraction with height for various values of — AN
-AN
Ns
Ce
-dNo
k
20
180. 226277
0.117626108
21. 1993155
1. 15617524
22
197.316142
. 118216356
23. 3259953
1. 17457412
24
212. 917967
. 119594076
25. 4637276
1. 19366808
26
227. 270255
. 121491305
27.6113599
1.21348565
28
240. 558398
. 123746115
29. 7681671
1.23406110
30
252. 929362
. 126255291
31.9336703
1. 25543336
32
264. 501627
. 128950180
34. 1075325
1. 27764560
34
275. 372099
. 131783550
36. 2895127
1. 30074523
36
285. 621054
. 134721962
38. 4794288
1. 32478398
38
295. 315731
. 137741207
40. 6771452
1.34981825
40
304. 513148
. 140823306
42. 8825481
1. 37590934
42
313.261483
. 143955014
45.0955611
1.40312414
44
321. 602888
. 147125889
47.3161107
1.43153539
46
329. 573439
. 150328075
49. 5441405
1. 46122250
48
337. 204713
. 153555418
51. 7796106
1.49227226
50
344. 524418
. 156803056
54. 0224815
1. 52477960
52
351. 557000
. 160067149
56. 2727266
1. 55884863
54
358. 324138
. 163344614
58. 5303179
1. 59459364
56
364. 845143
. 166633002
60. 7952415
1.63214058
58
371. 137293
. 169930326
63.0674811
1. 67162830
60
377. 216108
. 173234984
65. 3470266
1.71321044
62
383. 095581
. 176545680
67. 6338699
1. 75705732
64
388. 788373
. 179861358
69. 9280046
1. 80335830
66
394. 305974
. 183181171
72. 2294300
1.85232456
68
399. 658845
404. 856538
. 186504431
. 189830583
74. 5381454
76.8541525
1. 90419225
70
1. 95922635
72
409. 907798
. 193159183
79. 1774555
2.01772514
74..._
414. 820650
. 196489873
81. 5080567
2. 08002556
76
419. 602477
. 199822385
83. 8459677
2. 14650999
78
424. 260086
. 203156494
86. 1911914
2. 21761358
80
428. 799768
. 206492043
88. 5437400
2. 29383429
82
433. 227348
. 209828917
90. 9036251
2. 37574437
84
437. 548229
. 213167031
93. 2708570
2. 46400458
86
441. 767432
. 216506335
95. 6454475
2. 55938222
88
445. 889634
449. 919193
. 219846812
. 223188453
98. 0274147
100. 4167688
2. 66277367
90
2. 77523207
92
453. 860184
. 226531281
102. 8135290
2. 89800399
94
457. 716416
. 229875327
105. 2177108
3. 03257531
96..
461.491458
465. 188659
468.811163
. 233220637
. 236567271
.239915290
107. 6293319
110.0484114
112.4749663
3. 18073184
98
3. 34463902
100
3. 52694820
396 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
Table 9.29. Parameters for exponential refraction with height for various values of k
fc
N,
-AN
Cc
-dNo
1.0
1.2
1.3
1.4
1.5
0.0
217. 689023
275. 037959
312. 297111
339. 003316
359. 298283
375. 341242
388. 391792
399. 243407
408. 424907
416. 304322
423. 146728
429. 148472
434.458411
439. 191718
443. 438906
447. 272272
450. 750273
523. 299600
289. 036274
0.0
24. 6471681
33. 9367000
41. 7747176
48. 4839018
54. 2941700
59. 3759008
63. 8586055
67. 8426334
71.4070090
74. 6148487
77. 5171828
80. 1557288
82. 5649192
84. 7734613
86. 8054237
88.6811886
90. 4181120
135. 5109159
36. 6922523
0.0
. 120160519
. 131692114
. 143600133
. 154339490
. 163827653
. 172203063
. 179626805
. 186242834
. 192172034
. 197514185
. 202351472
. 206751820
. 210771674
. 214458304
. 217851443
. 220984823
. 223887193
. 299693586
. 135758874
0.0
26. 1576259
36. 2203304
44. 8459068
52. 3215987
1.6 ..-
58. 8629945
1.7
64.6349115
1.8
69. 7655765
1.9
74. 3562234
2.0
78. 4878451
2.1
82. 2260091
2.2
85. 6243634
2.3 ..
88. 7272277
2.4
91. 5715264
2.5
94. 1883108
2.6
96. 6038056
2.7
98. 8403838
2.8
100. 9172131
156. 829534
4/3
39. 2392391
9.3. Refractivity Tables
Tables 9.30 through 9.35 have been prepared on a digital computer to
assist the reader in making an easy determination of the refractivity, N,
value from pressure, temperature, and relative humidity data. Standard
pressure values of 700, 850, and 1000 mbar have been used for the tables.
The tables are at every 10 percent relative humidity and for every degree
centigrade ranging for —45 to +45 °C.
Interpolation for the A^ tables is accomplished in the following manner.
First, round the temperature to the nearest degree, then interpolate
linearly for the values of pressure, P, and relative humidity, RH, used.
No distortion is introduced by this method as the wet term, W (as de-
scribed in chapter 4), is a linear function of RH, and the dry term, D, is
a linear function of P.
Example:
and knowing
then
Given P = 980 mbar
RH = 47 percent
T = 22 °C
D = D(P, T)
N = N(P, T, RH)
W = W{T, RH),
Z)(850, 22) = A^(850, 22, 0) = 223.6 A^ units.
Z)(980, 22) = A^(980, 22, 0) = 223.6 (980/850)
257.8 N units.
REFRACTIVITY TABLES 397
Now interpolating for humidity,
N{850, 22, 50) = 280.3 N units.
Since
A^(850, 22, 50) - Z)(850, 22) = T^(22, 50),
thus
TF(22, 50) = 280.3 - 223.6 = 56.7 N units, and
TF(22, 47) = 56.7 (47/50) = 53.3 A^ units.
Adding
T^^(22, 47) to i)(980, 22) yields
A^(980, 22, 47) = D(980, 22) + IF(22,47)
= 257.8 + 53.3 = 311.1 N units.
Substitution directly into (1.20) and use of the Smithsonian Meteor-
ological Tables [2] for vapor pressure yields the identical result to four
significant figures.
A^(980, 22, 47) = 311.1 A^ units.
Ambient recording thermometers generally read to a tenth of a degree
centigrade. Rounding the temperature to the nearest whole degree
introduces an extraneous error into the interpolation process. At 45 °C,
1000 mbar, and at a relative humidity of 100 percent, this error will have
a maximum of 7.7 A^ units. However, these conditions are an extreme
rarity climatologically speaking, and at normal temperature ranges an
error of more than 2 N units is unlikely.
398 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
Table 9.30. Refractivity for a pressure of 700 mhar
Relative humidity
Temper-
ature
10
20
30
40
50
60
70
80
90
100
°C
-45
238.2
238.3
238.4
238.5
238.6
238.6
238.7
238.8
238.9
239.0
239.0
-44
237.2
237.3
237.4
237.5
237.6
237.6
237.7
237.8
237 9
238.0
238.1
-43
236.2
236.3
236.4
236.5
236.6
236.7
236.8
236.9
237.0
237.1
237.1
-42
235.2
235.3
235.4
235.5
235.6
235.7
235.8
235.9
236.0
236.1
236.2
-41
234.1
234.3
234.4
234.5
234.6
234.7
234.8
235.0
235.1
235.2
235.3
-40
233.1
233.3
233.4
233.5
233.7
233.8
233.9
234.0
234.2
234.3
234.4
-39
232.1
232.3
232.4
232.6
232.7
232.9
233.0
233.1
233.3
233.4
233.4
-38
231.1
231.3
231.5
231.6
231.8
231.9
232.1
232.2
232.4
232.6
232.7
-37
230.2
230.3
230.5
230.7
230.9
231.0
231.2
231.4
231.5
231.7
231.9
-36
229.2
229.4
229.6
229.8
230.0
230.1
230.3
230.5
230.7
230.9
231.1
-35
228.2
228.4
228.6
228.9
229.1
229.3
229.5
229.7
229.9
230.1
230.3
-34
227.3
227.5
227.7
228.0
228.2
228.4
228.6
228.9
229.1
229.3
229 5
-33
226.3
226.6
226.8
227.1
227.3
227.6
227.8
228.1
228.3
228.6
228.8
-32
225. 4
225.7
225.9
226.2
226.5
226.7
227.0
227.3
227.6
227.8
228.1
-31
224.5
224.8
225.1
225.3
225.6
225.9
226.2
226.5
226.8
227.1
227.4
-30
223.5
223.9
224.2
224.5
224.8
225.1
225.5
225.8
226.1
226.4
226.8
-29
222.6
223.0
223.3
223.7
224.0
224.4
224.7
225.1
225.4
225.8
226.1
-28
221.7
222.1
222.5
222.9
223.2
223.6
224.0
224.4
224.8
225.1
225.5
-27
220.8
221.2
221.6
222.1
222.5
222.9
223.3
223. 7
224.1
224.5
225.0
-26
219.9
220.4
220.8
221.3
221.7
222.2
222.6
223.1
223.5
224.0
224.4
-25
219.0
219.5
220.0
220.5
221.0
221.5
222.0
222.5
222.9
223.4
223.9
-24
218.2
218.7
219.2
219.7
220.3
220.8
221.3
221.9
222.4
222.9
223.5
-23
217.3
217.9
218.4
219.0
219.6
220.2
220.7
221.3
221.9
222.5
223.0
-22
216.4
217.0
217.7
218.3
218.9
219.5
220.2
220.8
221.4
222.0
222.7
-21
215.6
216.2
216.9
217.6
218.3
218.9
219.6
220.3
221.0
221.6
222.3
-20
214.7
215.4
216.2
216.9
217.6
218.4
219.1
219.8
220.6
221.3
222.0
-19
213.9
214.6
215.4
216.2
217.0
217.8
218.6
219.4
220.2
221.0
221.8
-18
213.0
213.9
214.7
215.6
216.4
217.3
218.1
219.0
219.9
220.7
221.6
-17
212.2
213.1
214.0
215.0
215.9
216.8
217.7
218.6
219.6
220.5
221.4
-16
211.4
212.4
213.4
214.3
215.3
216.3
217.3
218.3
219.3
220.3
221.3
-15
210.5
211.6
212.7
213.8
214.8
215.9
217.0
218.0
219.1
220.2
221.3
-14
209.7
210.9
212.0
213.2
214.3
215.5
216.7
217.8
219.0
220.1
221.3
-13
208.9
210.2
211.4
212.7
213.9
215.1
216.4
217.6
218.9
220.1
221.4
-12
208.1
209.5
210.8
212.1
213.5
214.8
216.1
217.5
218.8
220.2
221.5
-U
207.3
208.8
210.2
211.6
213.1
214.5
216.0
217.4
218.8
220.3
221.7
-10
206.5
208.1
209.6
211.2
212.7
214.3
215.8
217.4
218.9
220.4
222.0
-9
205.8
207.4
209.1
210.7
212.4
214.1
215.7
217.4
219.0
220.7
222.3
-8
205.0
206.8
208.5
210.3
212.1
213.9
215.7
217.4
219.2
221.0
222.8
-7
204.2
206.1
208.0
29.9
211.8
213.8
215.7
217.6
219.5
221.4
223.3
-6
203.4
205.5
207.5
29.6
211.6
213.7
215.7
217.8
219.8
221.9
223.9
-5
202.7
204.9
207.1
209.3
211.4
213.6
215.8
218.0
220.2
222.4
224.6
-4
201.9
204.3
206.6
209.0
211.3
213.7
216.0
218.3
220.7
223.0
225.4
-3
201.2
203.7
206.2
208.7
211.2
213.7
216.2
218.7
221.2
223.8
226.3
-2
200.4
203.1
205.8
208.5
211.2
213.8
216.5
219.2
221.9
224.6
227.3
-1
199.7
202.6
205.4
208.3
211.2
214.0
216.9
219.8
222.6
225.5
228.4
199.0
202.0
205,1
208.2
211.2
214.3
217.3
220.4
223.4
226.5
229.6
REFRACTIVITY TABLES
Table 9.31. RefractivUy for a pressure of 700 nibar.
399
Relative humidity
Temper-
ature
10
20
30
40
50
60
70
80
90
100
° C
199.0
202.0
205.1
208.2
211.2
214.3
217.3
220.4
223.4
226.5
229.0
1
198.2
201.5
204.8
208.0
211.3
214.6
217.8
221.1
224.4
227.6
230.9
2
197.5
201.0
204.5
208.0
211.5
214.9
218.4
221.9
225.4
228.9
232.3
3
196.8
200.5
204.2
207.9
211.7
215.4
219.1
222.8
226.5
230.2
233.9
4
196.1
200.1
204.0
208.0
211.9
215.9
219.8
223.8
227.7
231.7
235,6
5
195.4
199.6
203.8
208.0
212.2
216.5
220.7
224.9
229.1
233.3
237,5
6
194.7
199.2
203.7
208.1
212.6
217.1
221.6
226.1
230.5
235.0
239,5
7
194.0
198.8
203.5
208.3
213.1
217.8
222.6
227.4
232.1
236.9
241,7
8
193.3
198.4
203.4
208.5
213.6
218.7
223.7
228.8
233.9
238.9
244,0
9
192.6
198.0
203.4
208.8
214.2
219.6
224.9
230.3
235.7
241.1
246,5
10
191.9
197.7
203.4
209.1
214.8
220.5
226.3
232.0
237.7
243.4
249,1
11
191.3
197.3
203.4
209.5
215.6
221.6
227.7
233.8
239.8
245.9
252,0
12
190.6
197.0
203.5
209.9
216.4
222.8
229.2
235.7
242.1
248.6
255,0
13
189.9
196.8
203.6
210.4
217.3
224.1
230.9
237.7
244.6
251.4
258,2
14
189.3
196.5
203.7
211.0
218.2
225.5
232.7
239.9
247.2
254.4
261,7
15
188.6
196.3
204.0
211.6
219.3
227.0
234.6
242.3
250.0
257.6
265,3
16
188.0
196.1
204.2
212.3
220.4
228.6
236.7
244.8
252.9
261.1
269,2
17
187.3
195.9
204.5
213.1
221.7
230.3
238.9
247.5
256.1
264.7
273,3
18
186.7
195.8
204.9
213.9
223.0
232.1
241.2
250.3
259.4
268.5
277,6
19
186.0
195.6
205.3
214.9
224.5
234.1
243.7
253.3
262.9
272.6
282,2
20
185.4
195.6
205.7
215.9
226.0
236.2
246.4
256.5
266.7
276.9
287.0
20
184.8
195.5
206.2
217.0
227.7
238.4
249.2
259.9
270.6
281.4
292,1
22
184.1
195.5
206.8
218.1
229. 5
240.8
252.2
263.5
274.8
286.2
297,5
23
183.5
195.5
207.4
219.4
231.4
243.3
255.3
267.3
279.2
291.2
303,2
24
182.9
195.5
208.1
220.8
233.4
246.0
258.6
271.3
283.9
296.5
309,1
25
182.3
195.6
208.9
222.2
235.5
248.8
262.2
275.5
288.8
302.1
315.4
26
181.7
195.7
209.7
223.8
237.8
251.8
265.9
279.9
293.9
308.0
322.0
27
181.1
195.9
210.6
225.4
240.2
256.
269.8
284.6
299.3
314.1
328,9
28
180.5
196.0
211.6
227.2
242.7
258.3
273.9
289.5
305.0
320,6
336,2
29
179.9
196.3
212.7
229.0
245.4
261.8
278.2
294.6
311.0
327.4
343,8
30
179.3
196.5
213.8
231.0
248.3
265.5
282.8
300.0
317.3
334.5
351,8
31
178.7
196.8
215.0
233.1
251.3
269.4
287.6
305.7
323.8
342.0
360,1
32
178.1
197.2
216.3
235.3
254.4
273.5
292.6
311.7
330.7
349.8
368.9
33
177.5
197.6
217.6
237.7
257.7
277.8
297.8
317.9
337.9
258.0
378.1
34
176.9
198.0
219.1
240.1
261.2
282.3
303.4
324.4
345. 5
366.6
387.6
35
176.4
198.5
220.6
242.7
264.9
287.0
309.1
331.3
353.4
375.5
397.6
36
175.8
199.0
222.3
245.5
268.7
291.9
315.2
338.4
361.6
384.9
408.1
37
175.2
199.6
224.0
248.4
272.7
297.1
321.5
345.9
370.2
394.6
419.0
38
174.7
200,2
225.8
251.4
277.0
302.5
328.1
353.7
379.2
404.8
430.4
39
174.1
200.9
227.7
254.5
281.4
308.2
335.0
361.8
388.6
415,4
442,3
40
173.5
201.7
229.8
257.9
286.0
314.1
342.2
370.3
398.4
426,5
454.6
41
173.0
202.4
231.9
261.4
290.8
320.3
349.7
379.2
408.6
438.1
467,5
42
172.4
203.3
234.1
265.0
295.9
326.7
357.6
388.4
419.3
450.1
481,0
43
171.9
204.2
236.5
268.8
301.1
333.4
365.7
398.0
430.3
462.6
494,9
44
171.4
205.2
239.0
272.8
306.6
340.4
374.2
408.1
441.9
475.7
509.5
45
170.8
206.2
241.6
277.0
312.3
347.7
383.1
418.5
453.9
489.2
524,6
400 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
Table 9.32. Refractivity for a pressure of 850 mbar
Temper-
Relative humidity
ature
10
20
30
40
50
60
70
80
90
100
° C
-45
289.3
289.4
289.5
289.5
289.6
289.7
289.8
289.9
289.9
290.0
290.1
-44
288.0
288.1
288.2
288.3
288.4
288.5
288.6
288.7
288.7
288.8
288.9
-43
286.8
286.9
287.0
287.1
287.2
287.3
287.4
287.5
287.6
287.7
287.8
-42
285.5
285.6
285.8
285.9
286.0
286.1
286.2
286.3
286.4
286.5
286.6
-41
284.3
284.4
284.5
284.7
284.8
284.9
285.0
285.1
285.3
285.4
285.5
-40
283.1
283.2
283.4
283.5
283.6
283.7
283.9
284.0
284.1
284.3
284.4
-39
281.9
282.0
282.2
282.3
282.5
282.6
282.7
282.9
293.0
283.2
283.3
-38
280.7
280.8
281.0
281.2
281.3
281.5
281.6
281.8
281.9
282.1
282.3
-37
279.5
279.7
279.8
280.0
280.2
280.4
280.5
280.7
280.9
281.0
281.2
-36
278.3
278.5
278.7
278.9
279.1
279.3
279.4
279.6
279.8
280.0
280.2
-35
277.1
277.3
277.6
277.8
278.0
278.2
278.4
278.6
278.8
279.0
279.2
-34
276.0
276.2
276.4
276.7
276.9
277.1
277.3
277.6
277.8
278.0
278.2
-33
274.8
275.1
275.3
275.6
275.8
276.1
276.3
276.6
276.8
277.1
277.3
-32
273.7
274.0
274.2
274.5
274.8
275.0
275.3
275.6
275.9
276.1
276.4
-31
272.6
272.9
273.2
273.4
273.7
274.0
274.3
274.6
274.9
275.2
275.5
-30
271.4
271.8
272.1
272.4
272.7
273.0
273.4
273.7
274.0
274.3
274.7
-29
270.3
270.7
271.0
271.4
271.7
272.1
272.4
272.8
273.1
273.5
273.8
-28
269.2
269.6
270.0
270.4
270.8
271.1
271.5
271.9
272.3
272.7
273.0
-27
268.1
268.5
269.
269.4
269.8
270.2
270.6
271.0
271.4
271.9
272.3
-26
267.0
267.5
267.9
268.4
268.8
269.3
269.8
270.2
270.7
271.1
271.6
-25
266.0
266.5
266.9
267.4
267.9
268.4
268.9
269.4
269.9
270.4
270.9
-24
264.9
265.4
266.0
266.5
267.0
267.6
268.1
268.6
269.2
269.7
270.2
-23
263.8
264.4
265.0
265.6
266. 1
266.7
267.3
267.9
268.4
269.0
269.6
-22
262.8
263.4
264.0
264.7
265.3
265.9
266.5
267.2
267.8
268.4
269.0
-21
261.7
262.4
263.1
263.8
264.4
265.1
265.8
266.5
267.2
267.8
268.5
-20
260.7
261.4
262.2
262.9
263.6
264.4
265.1
265.8
266.6
267.3
268.0
-19
259.7
260.5
261.3
262.1
262.8
263.6
264.4
265.2
266.0
266.8
267.6
-18
258.7
259.5
260.4
261.2
262.1
262.9
236.8
264.6
265.5
266.4
267.2
-17
257.7
258.6
259.5
260.4
261.3
262.3
263.2
264.1
265.0
266.0
266.9
-16
256.7
257.6
258.6
259.6
260.6
261.6
262.6
263.6
264.6
265.6
266.6
-15
255.7
256.7
257.8
258.9
259.9
261.0
262.1
263.2
264.2
265.3
266.4
-14
254.7
255.8
257.0
258.1
259.3
260.4
261.6
262.8
263.9
265.1
266.2
-13
253.7
2.54.9
256.2
257.4
258.7
259.9
261.2
262.4
263.6
264.9
266.1
-12
252.7
254.1
255.4
256.7
258.1
259.4
260.7
262.1
263.4
264.8
266.1
-11
251.8
253.2
254.6
256.1
257.5
258.9
260.4
261.8
263.3
264.7
266.1
-10
250.8
252.3
253.9
255.4
257.0
258.5
260.1
261.6
263.2
264.7
266.2
-9
249.8
251.5
253.2
254.8
256.5
258.1
159.8
261.5
263.1
264.8
266.4
-8
248.9
250.7
252.5
254.2
256.0
257.8
259.6
261.4
263.1
264.9
266.7
-7
248.0
249.9
251.8
253.7
255.6
257.5
259.4
261.3
263.2
265.1
267.1
-6
247.0
249.1
251.1
253.2
255.2
257.3
259.3
261.4
263.4
265.4
267.5
-5
246.1
248.3
250.5
252.7
254.9
257.1
259.3
261.5
263.6
265.8
268.0
-4
245.2
247.5
249.9
252.2
254.6
256.9
259.3
261.6
264.0
266.3
268.6
-3
244.3
246.8
249.3
251.8
254.3
256.8
259.3
261.9
264.4
266.9
269.4
-2
243.4
246.1
248.8
251.4
254.1
256.8
259.5
262.2
264.8
267.5
270.2
-1
242.5
245.4
248.2
251.1
254.0
256.8
259.7
262.6
265.4
268.3
271.1
241.6
244.7
247.7
250.8
253.8
256.9
260.0
263.0
266.1
269.1
272.2
REFRACTIVITY TABLES
Table 9.33. Re fr activity for a pressure of 850 mbar
401
Relative humidity
Temper-
ature
10
20
30
40
50
60
70
80
90
100
° C
241.6
244.7
247.7
250.8
253.8
256.9
260.0
263.0
266.1
269.1
272.2
1
240.7
244.0
247.3
250.5
253.8
257.1
260.3
263.6
266.8
270.1
273.4
2
239.9
243.3
246.8
250.3
253.8
257.3
260.7
264.2
267.7
271.2
274.7
3
239.0
242.7
246.4
250.1
253.8
257.5
261.3
265.0
268.7
272.4
276.1
4
238.1
242.1
246.0
250.0
253.9
257.9
261.9
265.8
269.8
273.7
277.7
5
237.3
241.5
245.7
249.9
254.1
258.3
262.5
266.7
271.0
275.2
279.4
6
236.4
240.9
245.4
249.9
254.3
258.8
363.3
267.8
272.3
276.8
281.2
7
235.6
240.3
245.1
249.9
254.6
259.4
264.2
268.9
273.7
278.5
283.2
8
234.7
239.8
244.9
249.9
255.0
260.1
265.1
270.2
275.3
280.3
285.4
9
233.9
239.3
244.7
250.1
255.4
260.8
266.2
271.6
277.0
282.4
287.8
10
233.1
238.8
244.5
250.2
256.0
261.7
267.4
273.1
278.8
284.5
280.3
11
232.3
238.3
244.4
250.5
256.5
262.6
268.7
274.8
280.8
286.9
293.0
12
231.4
237.9
244.3
250.8
257.2
263.6
270.1
276. 5
283.0
289.4
295.9
13
230.6
237.5
244.3
251.1
258.0
264.8
271.6
278.4
285.3
292.1
298.9
14
229.8
237.1
244.3
251.5
258.8
266.0
283.3
280.5
287.7
295.0
302.2
15
229.0
236.7
244.4
252.0
259.7
267.4
275.0
282.7
290.4
298.1
305.7
16
228.2
236.4
244.5
252.6
260.7
268.8
277.0
285.1
293.2
301.3
309.5
17
227.4
236.0
244.6
253.2
261.8
270.4
279.0
287.6
296.2
304.8
313.4
18
226.7
235.8
244.9
253.9
263.0
272.1
281.2
290.3
299.4
308.5
317.6
19
225.9
235.5
245.1
254.7
264.4
274.0
283.6
293.2
302.8
212.4
322.0
20
225.1
235.3
245.4
255.6
265.8
275.9
286.1
296.3
306.4
316.6
326.7
21
224.4
235.1
245.8
256.6
267.3
278.0
288.8
299.5
310.2
321.0
331.7
22
223.6
234.9
246.3
257.9
268.9
280.3
291.6
302.9
314.3
325.6
337.0
23
222.8
234.8
246.8
258.7
270.7
272.7
294.6
306.6
318.6
330.5
242.5
24
222.1
234.7
247.3
260.0
272.6
285.2
297.8
310.4
323.1
335.7
348.3
25
221.3
234.7
248.0
261.3
274.6
287.9
301.2
314.5
327.8
341.1
354.5
26
220.6
234.6
248.7
262.7
276.7
280.9
304.8
318.8
332.9
346.9
360.9
27
219.9
234.7
249.4
264.2
279.0
293.8
308.6
323.4
338.1
352.9
367.7
28
219.1
234.7
250.3
265.8
281.4
297.0
312.6
328.1
343.7
359.3
374.8
28
218.4
234.8
251.2
267.6
284.0
300.4
316.8
333.2
349.6
365.9
382.3
30
217.7
234.9
252.2
269.4
286.7
303.9
321.2
338.4
355.7
372.9
390.2
31
217.0
235.1
253.3
271.4
289.6
307.7
325.8
344.0
362.1
380.3
398.4
32
216.3
235.3
254.4
273.5
292.6
311.7
330.7
249.8
368.9
388.0
407.1
33
215.6
235.6
255.7
275.7
295.8
315.8
335.9
355.9
376.0
296.0
416.1
34
214.9
235.9
257.0
278.1
299.1
320.2
341.2
362.3
383.4
404.5
425.5
35
214.2
236.3
258.4
280.5
302.7
324.8
346.9
369.0
391.2
413.3
435.4
36
213.5
236.7
259.9
283.2
305.4
329.6
352.8
376.1
399.3
422.5
445.8
37
212.8
237.2
261.5
285.9
310.3
334.7
359.0
383.4
407.8
432.2
456.5
38
212.1
237.7
263.2
288.8
314.4
340.0
365.5
391.1
416.7
422.2
467.8
39
211.4
238.2
265.0
291.9
318.7
345.5
372.2
399.1
425.9
452.7
479.6
40
210.7
238.8
267.0
295.1
323.2
351.3
379.4
407.5
435.6
463.7
491.8
41
210.1
239.5
269.0
298.4
327. 9
357.3
386.8
416.2
445.7
475.1
504.6
42
209.4
240.2
271.1
302.0
332.8
363.7
394.5
425.4
456.2
487.1
517.9
43
208.7
241.0
283.3
305.6
338.0
370.3
402.6
434.9
467.2
499.5
531.8
44
208.1
241.9
275.7
309.5
343.3
377.1
411.0
444.8
478.6
512.4
546.2
45
207.4
242.8
278.2
313.6
348.9
384.3
419.7
455.1
490.5
525.8
561.2
402 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
Table 9.34. Refractivity for a pressure of 1,000 mbar
Relative humidity
Temper-
ature
10
20
30
40
50
60
70
80
90
100
° C
-45
340.4
340.4
340.5
340.6
340.7
340.8
340.8
340.9
341.0
241.1
341.1
-44
338.9
339.0
339.0
339.1
339.2
339.2
339.4
339.5
339.6
339.7
339.7
-43
337.4
337.5
337.6
337.7
337.8
337.9
338.0
338.1
338.2
338.3
338.4
-42
335.9
336.0
336.1
336.3
336.4
336.5
336.6
336.7
336.8
336.9
337.0
-41
334.5
334.6
334.7
334.8
335.0
335.1
335.2
335.3
335.4
335.5
335.7
-40
333.0
333.2
333.3
333.4
333.6
333.7
333.8
334.0
334.1
334.2
334.3
-39
331.6
331.8
331.9
332.1
332.2
332.3
332.5
332.6
332.8
332.9
333.1
-38
330.2
330.4
330.5
330.7
330.8
331.0
331.2
331.3
331.5
331.6
331.8
-37
328.8
329.0
329.2
329.3
329.5
329.7
329.8
330.0
330.2
330.4
330.5
-36
327.4
327.6
327.8
328.0
328.2
328.4
328.6
328.7
328.9
329.1
329.3
-35
326.1
326.3
326.5
326.7
326.9
327.1
327.3
327.5
327.7
327.9
328.1
-34
324.7
324.9
325. 1
325.4
325. 6
325.8
326.0
326.3
326.5
326.7
326.9
-33
323.3
323.6
323.8
234.1
324.3
324.6
324.8
235.1
325.3
325.6
325.8
-32
322.0
322.3
322.6
322.8
323.1
323.3
323.6
323.9
324.2
324.4
324.7
-31
320.7
321.0
321.3
321.5
321.8
322.1
322.4
322.7
323.0
323.3
323.6
-30
319.3
319.7
320.0
320.3
320.6
320.9
321.3
321.6
321.9
322.2
322.6
-29
318.0
318.4
318.7
319.1
319.4
319.8
320.1
320.5
320.8
321.2
321.5
-28
316.7
317.1
317.5
317.9
318.3
318.6
319.0
319.4
319.8
320.2
320.5
-27
315.4
315.9
316.3
316.7
317.1
317.5
317.9
318.4
318.8
319.2
319.6
-26
314.2
314.6
315.1
315.5
316.0
316.4
316.9
317.3
317.8
318.2
318.7
-25
312.9
313.4
313.9
314.4
314.9
315.4
315.8
316.3
316.8
316.3
317.8
-24
311.6
312.2
312.7
313.2
313.8
314.3
314.8
315.4
315.9
316.4
317.0
-23
310.4
311.0
311.6
312.1
312.7
313.3
313.9
314.4
315.0
415.6
316.2
-22
309.2
309.8
310.4
311.0
311.7
312.9
313.5
314.2
314.2
314.8
315.4
-21
307.9
308.6
309.3
310.0
310.6
311.3
312.0
312.7
313.3
314.0
314.7
-20
306.7
307.6
308.2
308.9
309.6
310.4
311.1
311.8
312.6
313.3
314.0
-19
305.5
306.3
307.1
307.9
308.7
309.5
310.3
311.0
311.8
312.6
313.4
-18
304.3
305.2
306.0
306.9
307.7
308.6
309.4
310.3
311.1
312.0
312.9
-17
303.1
304.0
305.0
305.9
206.8
307.7
308.7
309.6
310.5
311.4
312.3
-16
301.9
302.9
303.9
304.9
305.9
306.9
307.9
308.9
309.9
310.9
311.9
-15
300.8
301.8
312.9
403.0
305.1
306.1
307.2
308.3
309.4
310.4
311.5
-14
299.6
300.8
301.9
303.1
304.2
305.4
306.5
307.7
308.9
310.0
311.2
-13
298.5
299.7
300.9
302.2
303.4
304.7
305.9
307.2
308.4
309.7
310.9
-12
297.3
298.7
300.0
301.3
302.7
304.0
305.3
306.7
308.0
309.4
310.7
-11
296.2
297.6
299.1
300.5
301.9
303.4
304.8
306.2
307.7
309.1
310.6
-10
295.1
296.6
298.1
299.7
301.2
302.8
304.3
305.9
307.4
309.0
310.5
-9
203.9
295.6
297.3
298.9
300.6
302.2
303.9
305.5
308.2
308.9
310.5
-8
292.8
294.6
296.4
298.2
299.9
301.7
303.5
305.3
307.1
308.8
310.6
-7
291.7
293.6
296.5
297.5
299.4
301.3
303.2
305.1
307.0
308.9
310.8
-6
290.6
292.7
294.7
296.8
298.8
300.9
302.9
305.0
307.0
309.0
311.1
-5
289.6
291.7
293.9
296.1
298.3
300.5
302.7
304.9
307.1
309.3
311.5
-4
288.5
290.8
293.2
295.5
297.9
300.2
302.5
304.9
307.2
309.6
311.9
-3
287.4
289.9
292.4
294.9
297.4
299.9
302.5
305.0
307.5
310.0
312.5
-2
296.3
289.0
291.7
294.4
297.1
299.8
302.4
305.1
307.8
310.5
313.2
-1
285.3
288.2
291.0
293.9
296.8
299.6
302.5
305.3
308.2
311.1
313.9
284.2
287.3
290.4
293.4
296.5
299.5
302.6
305.7
308.7
311.8
314.8
REFRACTIVITY TABLES
Table 9.35. Refractivity for a pressure of 1,000 mhar
403
Relative humidity
Temper-
ature
10
20
30
40
50
60
70
80
90
100
° C
284.2
287.3
290.4
293.4
296.5
299.5
302.6
305.7
308.7
311.8
314.8
1
283.2
286.5
289.7
293.0
296.3
299.6
302.8
306.1
309.3
312.6
315.9
2
282.2
285.7
289.1
292.6
269.1
299.6
303.1
306.6
310.0
313.5
317.0
3
281.2
284.9
288.6
292.3
296.0
299.7
303.4
307.1
310.9
314.6
318.3
4
280.1
284.1
288.1
292.0
296.0
299.9
303.9
307.8
311.8
315.7
319.7
5
279.1
283.3
287.6
291.8
296.0
300.2
304.4
308.6
213.8
317.0
321.2
6
278.1
282.6
287.1
291.6
296.1
300.5
305.0
309.5
314.0
318.5
323.0
7
277.1
281.9
286.7
291.4
296.2
301.0
305.7
310.5
315.3
320.0
324.8
8
276.2
281.2
286.3
291.4
296.7
301.5
306.6
311.6
316.7
321.8
326.8
9
275.2
280.6
285.9
291.3
296.7
302.1
307.5
213.9
318.3
323.6
329.0
10
274.2
279.9
285.6
291.4
297.1
302.8
308.5
314.2
320.0
325.7
331.4
11
273.2
279.3
285.4
291.5
297.5
303.6
309.7
315.7
321.8
327.9
334.0
12
272.3
278.7
285.2
291.6
298.0
304.5
310.9
317.4
323.8
330.3
336.7
13
271.3
278.2
285.0
291.8
298.7
305.5
312.3
319.1
326.0
332.8
339.6
14
270.4
277.6
284.9
292.1
299.3
306.6
313.8
321.1
328.3
335.5
342.8
15
269.4
277.1
274.7
292. 5
300.1
307.8
315.5
323.1
330.8
338.5
346.1
16
268.5
276.6
284.8
292.9
301.0
309.1
317.2
325.4
333.5
341.6
349.7
17
267.6
276.2
284.8
293.4
302.0
310.6
319.2
327. 8
336.4
344.9
353.5
18
266.7
275.8
284.9
293.9
303.0
312.1
321.2
330.3
339.4
348.5
357.6
19
265.8
275.4
285.0
294.6
304.2
313.8
323.4
333.1
342.7
352.3
361.9
20
264.8
275.0
285.2
295.3
305.5
315.8
325.8
336.0
346.1
356.3
366.5
21
363.9
274.7
285.4
296.2
306.9
317.6
328.4
339.1
349.8
360.6
371.3
22
263.1
274.4
285.7
297.1
308.4
319.7
331.1
342.4
353.7
365.1
376.4
23
262.2
274.1
286.1
298.1
310.0
322.0
334.0
345.9
357.9
369.8
381.8
24
261.3
273.9
286.5
299.1
311.8
324.4
337.0
349.6
362.3
374.9
387.5
25
260.4
273.7
287.0
300.3
313.6
327.0
340.3
353.6
366.9
380.2
393.5
26
259.5
273.6
287.6
301.6
315.7
329.7
343.7
357.8
371.8
385.8
399.8
27
258.7
273.5
288.2
303.0
317.8
332.6
347.4
362. 2
276.9
391.7
306.5
28
257.8
273.4
288.9
304.5
320.1
335.8
351.2
366.8
382.4
397.9
413.5
29
257.0
273.3
289.7
306.1
322.5
338.9
355.3
371.7
388.1
404.5
420.9
30
256.1
273.4
290.6
307.9
325.1
342.4
359.6
276.9
384.1
411.4
428.6
31
255.3
273.4
291.6
309.7
327.8
346.0
364.1
382.3
400.4
418.6
436.7
32
254.4
273.5
292.6
311.7
330.7
349.8
368.9
388.0
407.1
426.1
445.2
33
253.6
273.6
293.7
313.8
333.8
363. 9
373.9
394.0
414.0
434.1
454.1
34
252.8
273.8
294.9
316.0
337.0
358.1
379.2
444.3
421.3
442.4
463.5
35
251.9
274.1
296.2
318.3
340.5
362.6
384.7
406.8
429.0
451.1
473.2
36
251.1
274.4
297.6
320.8
344.1
367.3
390.5
413.7
437.0
460.2
384.4
37
250.3
274.7
299.1
323.5
347.8
372.2
396.6
421.0
445.3
469.7
494.1
38
249.5
275.1
300.7
326.2
351.8
377.4
403.0
428.5
454.1
479.7
505.2
39
248.7
275.5
302.3
329.2
256.0
382.8
409.6
436.4
463.2
490.1
516.9
40
247.9
276.0
304.1
332.2
360.4
388.5
416.6:
444.7
472.8
500.9
529.0
41
247.1
276.6
306.0
335.5
364.9
394.4
423.9
453. 3
482.8
512.2
541.7
42
246.3
277.2
308.1
338.9
369.8
400.6
431.5
462.3
493.2
524.0
544.9
43
245. 6
277.9
310.2
342.5
374.8
407.1
439.4
471.7
504.0
536.3
568.6
44
244.8
278.6
312.4
246.2
380.0
413.9
447.7
481.5
515. 3
549. 1
582.9
45
244.0
279.4
314.8
350.2
385.5
420.9
456.3
491.7
527.1
562.5
597.8
404 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
9.4. Glimatological Data of the Refractive Index
for the United States
Charts of 8-year mean A''o values for 0200 and 1400 local time are given
for the four seasons of the year on figures 9.1 to 9.8.
Selected stations illustrate the range of seasonal and diurnal variations
of N s in differing climatic regions across the country. These are plotted
on figures 9.9 to 9.14. Note the large annual and diurnal ranges of means
for humid subtropical Washington, D.C. (fig. 9.9), compared with the
modest variations exhibited by maritime-dominated Tatoosh Island
(fig. 9.14). The results of these analyses are consistent with those ob-
tained from climatic studies of the refractive index structure over North
America, described in section 4 of chapter 4. That is, large seasonal and
diurnal ranges are exhibited by continental-climate stations (Colorado
Springs) and small ranges by maritime-climate stations (Tatoosh Island).
Cumulative distribution curves for A^^, useful in radio ray bending pre-
dictions, are given in figures 9.15 to 9.20 for the example stations above.
Figures 9.21 and 9.22, standard deviation of N s, provide a measure of
the accuracy of the A^o charts of figures 9.1 to 9.8. As has been noted
in chapter 4, a number of climatic features are apparent on the standard
deviation maps. The climatic stability of various regions of the country,
for example, is reflected in these charts. Small standard deviations
characterize the maritime climate of the west coast. By comparison, the
strong air mass changes of wintertime synoptic patterns sweeping across
the southeastern United States are indicated by large standard devia-
tions in that region (fig. 9.21).
9.5. Statistical Prediction of Elevation Angle Error
The elevation angle error, e, as defined in chapter 3, can, like bending,
be predicted from surface refractivity, N s, by an equation of the form
e = mNs + i.
In tables 9.36 to 9.44 for a given height h, the coefficients m and t have
been determined by a statistical regression performed on the Standard
CRPL Sample, each having a unique N s-
This process is analogous to the statistical bending prediction method
described in chapter 3.
ELEVATION ANGLE ERROR
405
Figure 9.L Mean No, February 0200.
Figure 9.2. Mean No, February I4OO.
406 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
Figure 9.3. Mean No, May 0200.
Figure 9.4. Mean No, May 14OO.
ELEVATION ANGLE ERROR
407
Figure 9.5. Mean No, August 0200.
Figure 9.6. Mean No, August I4OO.
408 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
Figure 9.7. Mean No, November 0200.
Figure 9.8. Mean No, November I4OO.
ELEVATION ANGLE ERROR
409
1 1 1
0200 Lncal Time
390
380
3T0
360
350
330
520
310
300
290
280
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Hour of the Doy
Figure 9.9. Annual and diurnal cycles of Ns /or Washington, D.C.
410 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
O2OOL0C0I Time
y
_
^
/
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JAN KAR MAT JULY SEPT
Month of the Yeor
JM NAD HAY
JULY SEPT
Month of the Yeor
Ns
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igust
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y
oaoo 1200
Hour of the Doy
OeOO 1200 I6<
Hour of the Doy
Figure 9.10. Annual and diurnal cycles of Ns /or San Antonio, Tex.
ELEVATION ANGLE ERROR
411
1 1 1
0200 LdcoI Time
y
s
/
\
/
y
■-,
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Month of the Year
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Month of the Year
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0400 oeoo 1200 K
Hour of the Day
Figure 9.11. Annual and diurnal cycles of Ns /or Bismarck, N.Dak.
412
CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
1 1 1
1400 Local Time
\
/
\
s
/
\
/
— '
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—
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Month of the Yeor
SO
300
290
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260
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240 h
230
220
210
200,
1
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owo can 1200
Hour of the Day
2000 2400
ono 1200 It
Hour of the Doy
Figure 9.12. Annual and diurnal cycles of Ns /or Colorado Springs, Colo.
ELEVATION ANGLE ERROR
413
N^ '"" -
Month of the Year
Month of the Year
February
J
^
,__.
^
--■,
;^
^
— ■
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\
1
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—
oeoo 12
Hour of the Day
leoo
ONO 1200 1600
Hour of the Day
Figure 9.13. Annual and diurnal cycles of Ng/or Salt Lake City, Utah.
414
CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
1 1 1
0200 Local Tltne
y
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320
310
300
290
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Month of ttie Year
fiebruory
Ma>
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300
290
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0800 I2W K
Hour of the Doy
0800 1200 l(
Hour of the Day
Figure 9.14. Annual and diurnal cycles of Ns/or Tatoosh Island, Wash.
ELEVATION ANGLE ERROR
415
-
^
a
^
Moy
i
...
|X
—
- =0800 n=2l7 J
i 1
-■
t^
^
=1400 n=2l7 1
=2000 n=2l7 J
1 ' '
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>
^
^
1
\
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—
1
i
\
N
^
N
x'^$^
■4^^
;
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— = 0200 n=248
— = 0800 n=248
= 1400 n=248
— =2000 n=248-
90 I 9B|99.5| 99.99 0.01 | O.S| 2 10
I 5 20 40 60 80 95 99 99 9 0.1 I 5 20 40 60 80 95 99 999
P OF N, > ORDINATE VALUE
Figure 9.15. Cumulative probability distribution of Ns, Washington, D.C.
416
CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
370
350
330
310
290
2 70
I
Febnjoryl I
!
1
"*«
^^f
r
i 1 :
=0800 n=2Z6
1 r^'^!^ 1
1 h —
= 1400 n=Z26
=2000 n=226
•^; I i ! 1
1
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360
340
320
300
280
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99.5 99.99
0.1 I 5 20 40 60 80 95 99 99.9 0.1 I 5 20 4060 80 95 99 99.9
P OF N, >0RDINATE VALUE
Figure 9.16. Cumulative probability distribution of Ng, San Antonio, Tex.
ELEVATION ANGLE ERROR
417
R
t>
rt
1 1
._
0800 n"223
=1400 n=226l
=20O0 n=2?6 1
=»
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November
1 1
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=1400 n=240l
=2000 n=24oJ
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9
9
8
9
9.5
99
260
99.3 99.99 0.01
I
5 20 40 60 80 95 99 999 0.1 I 5 20 40 60 80 95 99 999
P OF N, > ORDINATE VALUE
Figure 9.17. Cumulative probability distribution of N9, Bismarck, N.Dak.
418
CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
!
1 1 1 1
February
Ml!
-J
-
0800 n=225
— - 1400 n = 226
2000 n = 226
s
1
-
i=>
^***^
^
[?=?
5
;
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-,
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-=
1
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1 1 1 1
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0800 n = 248
~~-$
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2000 n = 248
"
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■^
s
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s
^>
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->
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-
■
-■
5
tl 0,5 I Z 5 10 20 30 40 50 60 ?ll 80 90 95 9« 99 935
P of N,>Ordmote Value
05 12 5 10 2030«)50EOTOiO 9095 90!
P of N, > Ordmole Value
!
August
o
^
!;
^
0800 n = 248
-,N
s
^
^
2000 n = 248
^.\
J
S
K
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S
k
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Nc
ve
m
be
.J
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0800 n = 240
1400 n=240
2000 n = 240
L
-
3
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ife
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-
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I S K) ZO3O4OSOC0TD80 90 95
P of N, > Ordmote Value
2 5 10 20304050607090 9035 9!!
P of N, > Ordinate Value
Figure 9.18. Cumulative 'probability distribution of Ns, Colorado Springs, Colo.
Fe
br
ry
0200 n=226
0800 0=^226
ks
:^,
?:;
»
~1
1400 = 226
2000 = 226
"■
-
-
^
~~
s
^
::;
~^
-
^^
May
1
1
1
^5S
?5
^
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0200 = 248
0800 = 248
■•^
^^
"^
-
1400 = 248
2000 = 248
^
^■-.^
■•>,
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>
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"x
.
~''
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^
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ugu
t
300
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0800 0=248
"^.^
\
1400 n = 248
2000 n = 248
280
>
s
N
V
^
260
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S
^
^~
=*i
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240
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,.,
0.
31
9
1
a
3
5
7
9
9
e
a
95
99
230
99.99 0.01
0.1 I 5 20 40 60 80 95 99 99.9 01 I 5 20 40 60 80 95 99 99 9
P OF N, > ORDINATE VALUE
10
ve
n
»
0200 = 240
0800 = 240
'i
^
^
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2000 0=240
--
■"^
G^
i^
S)
^
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~
,_
V
5SH
'^'
->
V
^
v!
31
5
1
3
5
7
9
9
8
9
9.5
99
FiGURK 9.19. Cumulative probability distribution of Ns, Salt Lake City, Utah.
ELEVATION ANGLE ERROR
419
Febfuo
'»
II 1
III 1
-0200 n = 242
-
:
0800n = 255n
I400n = 255 j
'«^
^
^
_._
-2
a
XJ
■Tl
>«
^
^
"i;
3
^
r-TT
Moy
-
"
^
0800n = 248
1400 n = 248
^
•<
^
^
LJ
-2000
1
"1" I
^
k
\
\
O.oi I 0.5| 2 I 10 30 1 50 1 70 I 90 { 98 |99.S | 99.99 0.01 0.5 2 10 SO 50 70 90 98 99.5 99.99
0.1 r 5 20 40 60 SO 95 99 99.9 0.1 I 5 20 40 60 80 95 99 99.9
P OF N, > ORDINATE VALUE
Figure 9.20. Cumulative probability distribution of Ns, Tatoosh Island, Wash.
Figure 9.2 L Standard deviation of Nb, February 0200.
420 CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
Figure 9.22. Standard deviation of Ns, August 0200.
Tables 9.36 to 9.44
These tables show values of m and i in the regression equation
e = mN s + ^, where:
e is the elevation angle error in milliradians to the height h,
N s is the value of the radio refractivity at the earth's surface,
m the change in elevation angle error per unit change in N s,
t is the zero intercept,
r is the correlation coefficient,
SE is the standard error of prediction using the regression line,
00 is the initial elevation angle in milliradians.
Table 9.36. Variables in the statistical method of elevation angle error for h = 0.1 km
do
T
m
I
SE
0.0
0. 2637
.2760
.2855
.3017
.1609
.1777
.1805
.1824
. 1815
.1844
0. 0243
.0131
.0083
.0037
.0027
.0012
.0005
.0002
.0001
.0001
-4. 3954
-2. 1055
-1.2185
-. 4741
-.2834
-.1175
-.0426
-.0227
-.0110
-. 0055
3. 4429
1.0
2.0
1.7712
1.0837
5.0
0. 4561
10.0
.6501
20.0
.2706
52.4
.0998
100.0
.0518
200.0
.0256
400.0
.0123
TABLES 9.36 TO 9.44
421
Table 9.37. Variables in the statistical method of elevation angle error for h = 0.2 km
So
r
m
I
SE
0.0
0. 2739
.2838
.2927
.3146
. 1919
.2125
.2168
.2175
.2170
.2170
0. 0310
.0191
.0132
.0065
.0044
.0021
.0008
.0004
.0002
.0001
-5. 5364
-3. 0776
-1.9685
-. 8506
-.4623
-.1941
-.0705
-.0367
-.0179
-.0085
4. 2249
1.0
2.0
2. 4984
1.6702
5.0
10. 0-
.7628
.8906
20.0
.3757
52.4
. 1385
100.0
.0721
200.0
.0356
400.0
.0170
Table 9.38. Variables in the statistical method of elevation angle error for h = 0.5 km
eo
r
m
I
SE
0.0
1.0
2.0
0. 3255
.3457
.3661
.4197
.2887
.3304
.3418
.3430
.3439
.3408
0. 0430
.0289
.0216
.0125
.0083
.0041
.0016
.0008
.0004
.0002
-8. 0534
-5. 0198
-3. 5807
-1.9306
-1.0741
-. 4899
-. 1803
-.0934
-. 0463
-.0218
4. 8023
3.0131
2. 1131
5.0
1.0427
10.0
1.0693
20.0-.-
.4568
52.4
100.0
200.0- .. -
.1679
.0872
.0431
400.0
.0208
Table 9.39. Variables in the statistical method of elevation angle error for h = 1.0 km
Bod
r
m
I
SE
0.0
0. 3371
.3772
.4180
.5150
.3983
.4831
.5129
. 5167
.5177
.5180
0. 0472
.0339
.0270
.0179
.0123
.0066
.0026
.0014
.0007
.0003
-8. 2579
-5. 5583
-4. 2916
-2. 7591
-1.6158
-.8154
-. 3070
-. 1595
-. 0787
-.0379
5. 1154
1.0
2.0
3. 2281
2. 2775
5.0
1. 1590
10.0
1. 1044
20.0
.4703
52.4
.1714
100.0
.0890
200.0- -.
.0440
400.0
.0211
Table 9.40. Variables in the statistical method of elevation angle error for h = 2.0 km
Oo
r
m
I
SE
0.0
0. 3795
.4413
.5020
.6351
.5642
.7007
. 7531
.7608
. 7633
. 7643
0. 0567
.0427
.0355
.0258
.0188
.0113
.0047
.0025
.0012
.0006
-9. 8908
-7. 0696
-5. 7509
-4. 1251
-2. 7184
-1.5650
-.6256
-.3279
-. 1627
-. 0784
5. 3659
1.0
2.0
3.3704
2. 3735
5.0
1.2198
10.0
1. 0773
20.0
.4499
52.4
.1615
100.0
.0835
200.0
.0412
400.0 -.
.0198
422
CHARTS, GRAPHS, TABLES, AND COMPUTATIONS
Table 9.41. Variables in. the statistical method of elevation angle error for h = 5.0 km
00
r
m
I
SE
0.0
0. 4324
.5186
.5983
.7543
.7424
.8789
.9248
.9316
.9336
.9339
0. 0704
.0554
.0477
.0372
.0290
.0194
.0089
.0048
.0024
.0012
-12.0857
-9. 0682
-7.6679
-5. 9085
-4. 3100
-2. 7570
-1.1721
-.6154
-.3043
-. 1460
5. 6994
1.0
2.0
3. 5425
2. 4766
5.0
1. 2570
10.0
1 0255
20.0
.4118
52.4...
. 1431
100.0
.0733
200.0
.0360
400.0
.0173
Table 9.42. Variables in the statistical method of elevation angle error for h = 10.0 km
do
r
m
I
SE
0.0
0. 4617
.5602
.6475
.8057
.8125
.9301
.9679
.9742
.9763
.9766
0. 0797
.0637
.0555
.0443
.0353
.0245
.0118
.0065
.0032
.0016
-13.2953
-10.0839
-8. 5935
-6. 6912
-4. 9338
-3. 1637
-1.3002
-. 6607
-.3208
-. 1533
5. 9448
1.0
2.0
3.6560
2.5360
5.0
1. 2652
10.0
.9913
20.0
52.4
.3779
.1195
100. 0.
.0584
200.0.
.0281
400.0
.0135
Table 9.43. Variables in the statistical method of elevation angle error for h = SO.O km
So
r
TO
I
SE
0.0
1.0
2. 0--
0. 4805
.5866
.6780
.8351
.8497
.9538
.9861
.9917
.9934
.9939
0. 0874
.0703
.0616
.0495
.0397
.0278
.0136
.0076
.0038
.0018
-14.0318
-10.5942
-8. 9972
-6. 9273
-5. 0189
-3. 0650
-1.0682
-.4672
-.2067
-.0947
6. 1895
3. 7658
2. 5905
5.0
1. 2665
10.0
.9637
20.0
.3423
52.4
.0899
100.0 .
.0384
200.0
.0172
400.0
.0080
Table 9.44. Variables in the statistical method of elevation angle error for h = 70.0 km
Bo
r
TO
I
SE
0.0
0. 4999
.6137
.0783
.8617
.8811
.9701
.9948
.9983
.9992
.9995
0. 0985
.0794
.0698
.0563
.0451
.0317
.0158
.0089
.0045
.0022
-15.2202
-11.3649
-9. 5814
-7. 2363
-5.0975
-2.9121
-.8032
-. 2595
-.0822
-.0319
6.6214
1.0
2.0
3. 9661
2. 6984
5.0 --
1. 2867
10.0
.9476
20.0
.3102
52.4
.0630
100.0
.0202
200.0
.0070
400.0
.0028
REFERENCES 423
9.6. References
[1] Bean, B. R., and G. D. Thayer (1959), On models of the atmospheric radio refrac-
tive index, Proc. IRE 47, No. 5, 740-755.
[2] Smithsonian Meteorological Tables (1951), Table 63, Sixth Revised Ed. (Wash-
ington, D.C.).
Subject Index
A-unit (potential refractivity), 16-20, 78-79, 166-170, 189-194, 197, 211, 213-227, 379
Absolute humidity, 270, 272, 275, 277-280, 286-290
Absorption coefficient (see also Van Vleck's absorption formulas), 272
Adiabatic refractivity decrease, 258-262
dry, 261
wet, 262
Adiabatic temperature lapse rate, 133, 258
Advection, 132, 201, 205, 213, 319
definition, 132
relative to duct formation, 133, 135
Air masses, 79, 102, 114, 119, 123, 163-164, 167-169, 173-176, 179-181, 191, 195-
197, 201, 213, 221-222
refractive characteristics of, 163, 169-170, 174-175, 211
Angle of penetration (critical angle), 137, 140, 142, 376
Apparent height, 360
Apparent range, 51, 335, 357-360
ARDC Standard Atmosphere, 64
Attenuation of radio waves, 269-309
by fog, 303
by gases, 270-290
by hail, 302-303
by rain {see also Climatology), 292-297
in clouds, 291
B
B-unit, 14-15, 17-20, 189-191, 195
Barium fluoride strip, 26
Bending of a radio ray (see also Refraction)
definition, 49
methods of computing and predicting, 53-81, 375-393
Bi-exponential model
for absorption, 283, 307
for refractivity, 311-322
Birnbaum refractometer, 31, 33-34
Bright band, 303
C
Caustic, 148
Climatic types, relation to Na, 102-104
Climatology, 89-172
of ground based radio ducts, 42-43, 132-145
of refractivity, A^, near the ground (No), 89-109, 111-128, 404-420
of rainfall attenuation, 297-302
Clouds, attenuation in, 291
Cold front (see Frontal zone)
Composite parameter, 238
Continental cross sections of No, 122-128
Continental arctic air mass (see also Continental polar air mass), 102, 316
Continental polar air mass, 79, 123, 163-164, 169, 174-176, 179-180, 196-197,
221-222, 319
Continental tropical air mass, 79, 167-169, 175, 179
Convergence
centers of, 182, 240-241
influencing atmospheric refractivity, 182-185, 240
Core climate, 129
425
426 SUBJECT INDEX
Correlation of field strength with meteorological parameters, 105, 174, 229, 233-266
Crain refractometer, 31-33
Cross-over height, 248-249, 263-264
CRPL Reference Atmosphere {see Exponential atmosphere model for refractivity)
CRPL Standard A^-profile Sample, 335, 338-353, 355-356
Curvature of radio ray, 13, 42, 56, 264, 360-361
D
Deam refractometer, 36-37
Delta-A'^ (A N) {see also Refractivity, vertical gradient), 15-16, 61, 63-66, 73-76,
106, 129-131, 139, 229-230, 342, 363, 365, 371, 373, 389, 394-396
as a field strength predictor, 234-238, 240, 243-244, 246, 264-266
usage for 24-hour horizontal changes, 201-204, 209-211
Departures-from-normal method of computing refraction, 77-79, 167-168, 181, 190-
191, 375, 378-379, 381, 392-393
Dielectric constant of air {see also Refractive index), 2-8, 12
Dispersion, 5, 8
Diurnal index (diurnal ratio), 129-130
Diurnal range
of field strength, 105, 235-236
of A^o, 111-121, 129-130, 204
Divergence
centers of, 182, 240-241
influencing atmospheric refractivity, 182-185, 240
Drop size distribution, 292, 294-297
Dropsonde, 36
Dry term
of atmospheric absorption, 283
of the Smith- Weintraub equation, 30, 91, 93-94, 136, 174-175, 311-322
scale heights of (see Scale height)
surface values of, 93-94, 311-312, 320
Ducting, 42-45, 52, 89, 132-163, 241-246, 248, 324, 328, 331-333, 376
E
Effective earth, 13-15, 51, 56-62, 64-65, 106, 189, 195, 250, 264, 359-362
curvature, 13, 56-57, 62, 106, 195, 361
method of calculating refraction, 56-59
radius, 13-15, 51, 56-57, 59-62, 64-65, 106, 189, 195, 250, 264, 359-362
Elevated layer, 241, 244-258, 324-329, 331
reflection coefficient of, 252-257
Elevation angle (see also Refraction)
approximation for, 82-83
definition, 49-50
errors, 49-50, 178, 333-335, 340-342, 344-345, 348-350, 356, 404, 420-422
prediction of, 174, 334, 340-342, 344-345, 348-350, 356, 404, 420-422
Epoch chart, 221-227
Equation of continuity, 183
Equivalent gradient, 236-239, 243, 246, 264-266
as a field strength predictor, 236-239, 243, 246, 265
method of calculation, 264
Exponential atmosphere models
for absorption, 283
for refractivity, 15-20, 56, 61, 63-80, 92-93, 106, 110, 149, 153-156, 164, 166-
167, 174, 176-177, 186, 189-190, 192-194, 244-245, 261-263, 311-322,
329-332, 357, 362, 375-378, 385-389, 393-396
in radiosonde correction process, 40-41
method of computing refraction, 375-377, 385-388
Eta {t)) parameter, 238-239
F
Fading (see also Field Strength), 146-163, 223
caused by ducting, 146-163
definitions, 147
due to rainfall attenuation, 298-302
Field strength
prediction from radio meteorological parameters, 89, 105-106, 109, 174, 177-180,
229-236, 238-251, 253-258, 263-266
SUBJECT INDEX 427
Fog attenuation, 303
Four-thirds earth (see also Effective earth), 14-15, 20, 51, 57, 59-60, 62, 66-75,
247, 360
Fresnel discontinuity value, 252, 257
Frontal zone, influence on refractivity, 176-177, 187-193, 195-224, 323
G
Geometric range, 335, 357
Gradient of refractive index (see Refractive index)
Gradient of refractivity (see Refractivity)
regression expressions for, 363-364, 372
Graphical method of computing refraction, 80-81, 375, 380-381
Ground-based radio ducts (see also Ducting), 42-44, 132-163, 324, 328, 333, 375
frequency of occurrence, 140-141
thickness of, 140, 143-145
Ground-based superrefractive layers, 250
H
//-parameters, 265
Hail attenuation, 302-303
Half-length, 150-154, 160
Hay refractometer, 36-37
Height errors, 356-357, 360-373
definition, 356, 360
regression equations for, 364-372
High angle approximation for computing refraction, 53-54
Horizontal homogeniety, assumption of, 49, 52, 163, 322
Horizontal inhomogenieties, 52, 322-333, 356
Horizontal ray, 329
I
ICAO Standard Atmosphere, 9, 11, 312-314
Initial gradient of refractivity {see Delta-A'^ and Refractivity)
Intensity factor, 273-274
Inversions (see Ducting and Elevated layers)
K
ii:-unit, 180, 182, 239-240
L
Lag times, 29, 38-45
in electrical thermometers, 26
of relative humidity sensor in radiosondes, 39, 41-45
of temperature sensor in radiosondes, 39, 41-45
Land breeze, relative to duct formation, 134-135, 323
Level of nondivergence, 183
Lifting condensation level, 260
Line widths of atmospheric gaseous absorption (see also Attenuation), 272-275
Lithium chloride humidity strip, 26, 28-29, 39, 45
M
il/-parameter {see Thermal stability)
Macroscale fluctuation of refractive index, 173, 223
Maritime polar air mass, 79, 114, 169, 179, 197
Maritime tropical air mass, 79, 102, 119, 125-126, 164, 167, 169, 174, 177, 179-181,
201, 213, 221-222, 319
Maximum wavelengths trapped by a duct, 141-143
Mesocale fluctuation of refractive index, 173, 185, 223
Microscale fluctuation of refractive index, 173
Mixing ratio, 260
Models
of atmospheric absorption structure, 283
of atmospheres, 9, 11, 16-17, 64, 92-93, 180, 186, 312-314
of refractivity structure, 13-20, 40-41, 51, 56-80, 92-93, 106, 110, 149, 153-156,
164, 166-170, 174, 176-177, 180, 186, 189-195, 197, 213-227, 244-245, 250,
261-264, 311-322, 329, 331-332, 357, 360-362, 375-378, 381, 385-389,
393-396
428 SUBJECT INDEX
Modified effective earth's radius method of computing refraction, 59-65
Modified index of refraction (M-unit), 14-15, 17-20, 193, 195, 324
N
A'^-unit (see Refractivity)
NACA Standard Atmosphere, 16-17, 92-93, 186
Nocturnal duct formation (see also Ground-based radio ducts), 133, 135
Nonpolar gases, dielectric constant of, 2-3
O
Optical depth, 304
Outage time, 298-301
Oxygen
absorption properties of (see Attenuation by gases)
refractivity at STP, 9
P
Permeability of air, 3-4
Pi-factors (turbulence parameters), 262-266
Polar air mass (see Continental polar air mass)
Polar front (see also Frontal zone), 195-197, 213
Polar wave, 193, 195-197, 213
Polar gases, dielectric constant of, 2-3
Potential refractive modulus (<p-unit), 17, 19-20, 166, 193
Pound oscillator (see Deam refractometer)
Precipitation rate, 292
Profile (gradient) classification (see also Air masses and Refractivity), 248-250
Profiles of refractive index (see Refractive index, vertical profile of)
Profiles of refractivity (see Refractivity, vertical profile of)
Psychrometric equation, 12, 24
R
Radar
applications involving (see also Rain attenuation), 49, 51, 56, 59, 61, 77, 340, 357
anomalous propagation in (see Ducting)
Radiation, relative to duct formation, 133
Radiative transfer equation, 304
Radio
duct (see Ducting)
holes (see Ducting)
horizon
equation for, 59
in ducting, 146-148, 152, 155, 157
refractive index (see Refractive index)
Radiosonde (RAOB), 10-11, 26-30, 36, 38-43, 53-55, 59, 61, 70-72, 74-75, 79, 110-
111, 135, 138-140, 144, 179, 181, 194, 205, 211, 213, 243, 246, 247-249, 251, 253,
255, 258, 264, 280, 283, 326, 363, 375
sensors in, 26-27, 29-30, 39-45
Rain attenuation, 292-297
Range
errors (radio), 12, 174, 335-342, 345, 350-353, 356
definitions, 335-336
prediction of, 341, 345, 350-353
total, 335-337
of a radio ray, 51, 335-336, 357-361
Ray tracing (see also Refraction), 52, 132, 135-136, 148, 322-323
limitations of, 52, 132
Rayleigh-Jeans law, 304-305
Reflection coefficient (of an elevated layer), 185, 252-257
models of, 252-257
Refraction (angular) of a radio ray, 14-15, 19-20, 49-87, 163-170, 178-181, 190, 192,
236, 266, 311, 319-323, 326-337, 339, 341-350, 357, 375-396
definition, 49
derivation in terms of refractive index, 87
SUBJECT INDEX 429
Refractive index, atmospheric radio (see also Refractivity)
applications of, 12-14, 31-34, 36, 49-87, 136-138, 150-152, 165-166, 174, 223,
231-266, 311, 319-323, 326-329, 335-336, 357, 359-362, 394
definition, 23
dispersion, 5, 8
expressions for, 3-7
general discussion of, 1-8, 16-17, 19-20, 49, 89-90, 92, 103, 109-110, 125, 129,
133, 135, 163-166, 169-170, 173-175, 178, 185-195, 211, 222, 311, 314, 332-
333, 342-343, 356, 404
vertical gradient of (see also Delta-A^), 13-14, 19, 36, 38, 52, 56-57, 64, 77, 106-107,
132, 138-140, 142, 180, 182, 189, 195, 236, 240-241, 248, 264, 319, 357, 361-364
vertical profile of, 14-15, 40, 42-45, 52, 59, 62, 135, 173-176, 179-180, 182, 187,
211, 222, 229, 240, 247-252, 254, 257-258, 331-332, 334, 339-342, 363, 366
Refractivity (see also Refractive index)
applications of, 12-13, 19-20, 49-87, 105-107, 136-170, 178-181, 190, 229-266,
319-373, 375-396
as a function of height (see Models)
definition, 4, 90, 357
diurnal range (see Diurnal range)
expressions for, 4-7
constants in, 4-11
errors in, 9-12
measurement of, 23-45
accuracy and errors in, 24-32, 34-36, 38-45
near the ground (surface values), 15-20, 36, 42-43, 53-54, 58, 61, 63-80, 89-163,
176-182, 186-187, 192, 194, 197-205, 223, 229-230, 233-237, 239, 243-246,
258, 261-266, 311-312, 319-321, 333-356, 361-363, 365-367, 369-370, 373,
375-388, 392-396, 404-422
of atmospheric constituents
carbon dioxide, 5
dry air (see also Dry term), 9
oxygen, 9
water vapor (see also Wet term), 9
potential (see ^4 -unit)
scale heights of (see Scale height)
synoptic variation of (see Frontal zone)
vertical gradient of (see also Delta-AA), 9-12, 15, 20, 33, 36, 42-43, 61, 63-66,
73-77, 79, 106, 114, 123, 129-131, 136, 143, 145, 147-148, 150, 152, 166,
168, 229-230, 234-241, 243-244, 246, 248, 250, 264-266, 342, 362-363,
365-367, 370-371, 373, 375, 378, 389, 394-396
vertical profile of (see also CRPL Standard N-profile Sample), 15, 17-19, 42-45,
54-55, 57, 59-68, 70-71, 73, 75, 77-79, HI, 115, 117, 122, 129, 135, 138-
139, 146, 162-164, 167-170, 174-177, 194, 211-227, 235, 244-245, 263, 266,
321, 329, 336-339, 341, 344, 375, 381
Refractometer, 8, 23, 31-38, 334
Resonant frequencies (absorption lines)
of oxygen, 270-275, 280
of water vapor, 270-275, 280
Rocket Panel data, 64
S
Satellite telecommunications, 65, 175
Saturation vapor pressure, exponential assumption for, 260
Scale height
of absorption, 283
of refractivity, 15-17, 92-93, 108, 186, 261-263, 311-312, 314-322
of dry term, 93, 261, 263, 311-3)2, 314-322
of wet term, 93, 262-263, 311-312, 314-322
Scattering by atmospheric constituents (see also Attenuation) 185, 247-248, 255,
269-270, 292
Schulkin's method of computing refraction, 54-55, 80, 179, 375-376, 390-391
Sea breeze, relative to duct formation, 134-135, 323
Seasonal index (seasonal ratio), 129-130
430 SUBJECT INDEX
Seasonal range
of field strength, 174
of No {see also Climatology), 111-121, 129-130
Shadow zones, 148-150, 152-155, 157, 159-160
Slant range, 335, 360
Snell's law in polar co-ordinates, 49, 84-87, 136-137, 165, 326
derivation, 84-87
Snow attenuation, 302
Space cross sections of refractive index, 211-217, 221
Sprung's formula, 12
Stability
atmospheric, 238-241, 258, 260, 262-263, 266
thermal (see Thermal stability)
Standard atmosphere, 9, 11, 16-17, 64, 92-93, 180, 186, 312-314
Statistical method of computing refraction, 54, 335, 342-350, 379-380, 382-384
Stratopause, appoximate location, 313
Stratosphere
approximate extent, 313
meteorological characteristics of, 313
Subrefraction, 133, 248-249, 329, 332-333
Subsidence, relative to duct formation, 133-135, 180, 182, 185
Superior air mass, 79, 169
Superrefraction {see also Ducting), 133-135, 146-148, 156, 169, 182, 248, 250, 328-329,
332-333
Surface duct (see Ground-based radio ducts)
Synoptic illustration of A'' change (see Frontal zone)
T
Temperature dependence
of gaseous absorption (see Van Vleck's absorption formulas)
of refractivity, 1-8
Thermal noise temperature, 304-307
definition, 305
Thermal stability, 239-241, 262, 265
Tilted elevated layer, 249-251
Time cross sections of refractive index, 213, 217-221
Tracking system, radio, 334
Transhorizon propagation mechanisms, 248
Transition zone (see Frontal zone)
Trapping of a radio ray (see Ducting)
Tropopause, approximate location, 313
Troposphere
approximate extent, 313
meteorological characteristics of, 313
Turbulence
atmospheric, 260
parameter (see Pi-factors)
U
Unstratified layer, 248, 250-251
Van Vleck's absorption formulas, 270, 272-275
Velocity of radio propagation, 56
Vertical
gradient of A^ (see Refractivity)
profile of A'^ (see Refractivity)
ray, 329
Vetter refractometer, 31, 34-35
W
Warm front (see Frontal zone)
Water vapor
absorption properties of (see Attenuation by gases)
an average refractivity value of, 9
density (see absolute humidity)
dependence of refractivity, 1-8
SUBJECT INDEX 431
Weather influence on refractivity {see Frontal zone)
Wet term
of atmospheric absorption, 283
of the Smith-Weintraub equation, 30, 91, 93-94, 136, 174-175, 311-322
scale heights of {see Scale height)
surface values of, 93-94, 311-312, 320
Wind velocity
influence on atmospheric stability, 240-241
vertical component of, 182-184
Wiresonde, 326
Author Index
Abbott, R. 270, 274, 288, 308
Abild, V. B. 268
Adey, A. W. 46
Airy, G. B. 148, 172
Akita, K. I. 30, 45
Anderson, L. J. 22, 55, 80, 88, 171, 227,
268
Anderson, W. L. 374
Anway, A. C. 347, 374
Armstrong, H. L. 21
Arnold, E. 268
Artman, J. O. 273, 308
Arvola, W. A. 180, 182, 226, 227, 240,
266
Atwood, S. S. 8, 22, 296-297, 309
aufm Kampe, H. J. 291, 308
Autler, S. H. 273, 275, 308
Averbach, B. L. 46
B
Badcoe, S. R. 268
Barghausen, A. F. 374
Barrel), H. 5, 21
Barsis, A. P. 267
Battaglia, A. 8, 21
Battan, L. J. 291, 308
Bauer, J. R. 22, 255, 268, 357, 374
Bean, B. R. 22, 29, 46, 54, 87-88, 170-
172, 180, 225-228, 243, 266-268,
270, 274, 288, 308-309, 342, 363,
373-374, 377, 423
Becker, G. B. 273, 275, 308
Beckmann, P. 357, 374
Bennett, C. A. 374
Best, A. C. 309
Beyers, N. J. 374
Birnbaum, G. 5-7, 21-22, 31, 33-34, 46-
47, 273, 308
Boithias, L. 236, 267
Bonavoglia, L. 230, 266
Booker, H. G. 87, 171, 227, 267
Born, M. 21
Boudouris, G. 8, 21
Brekhovskikh, L. M. 254, 268
Bremmer, H. 136, 171
Brooks, C. E. P. 374
Buckley, F. 5, 21
Bunker, A. F. 39, 47
Burrows, G. R. 8, 22, 56, 88, 106, 171,
175, 225, 296-297, 309, 374
Bussev, H. K. 46, 297, 302, 308
Byers, H. R. 167, 172
Gaboon, B. A. 54, 88, 243, 266, 308, 363,
374
Carruthers, N. 374
Chatterjee, S. K. 6-7, 21
Clarke, L. C. 39, 47
Cline, D. E. 46
Clinger, A. H. 30, 46
Cole, C. F., Jr. 36, 47
Condron, T. D. 373
Cowan, L. W. 171
Craig, R. A. 8, 22, 267, 373
Grain, C. M. 6, 8, 21-22, 31-33, 46-47
Crawford, A. B. 8, 22, 225, 238, 256, 258,
267
Crozier, A. L. 30, 46
D
Davidson, D. 308
Debye, P. 2, 4, 21
Deam, A. P. 36-37, 46-47
Doherty, L. H. 148, 172
Donaldson, R. J., Jr. 308
Dubin, M. 64, 88
du Castel, F. 171, 253, 267-268
Dunmore, F. W. 47
Dutton, E. J. 29, 46, 88, 172, 228, 309
East, T. W. R. 274, 276, 291, 308
Edwards, H. D. 280, 309
Englund, C. R. 8, 22, 225
Essen, L. 5-6, 8, 10-12, 21-22
Evans, H. W. 295, 299-302, 309
Fannin, B. M. 172, 179, 226
Fehlhaber, L. 22, 171, 268
Ferrell, E. B. 8, 21, 56, 88, 106, 171, 225,
374
Flavell, R. G. 180, 182, 227, 240, 267
Franklin, N. L. 374
Eraser, 1). W. 46
Freehafer, J. E. 87, 374
Freethey, F. E. 46
Friis, H. T. 238, 256, 258, 267
Froome, K. D. 5-6, 8, 10-12, 21
Gallet, R. M. 22, 171
( larfinkel, S. 47
(ierson, N. C. 170, 174-176, 224
(;h(.sh, S. \. 280, 309
Gordon, J. P. 273, 308
Gordon, \V. E. 227, 267
433
434
AUTHOR INDEX
Gordy, W. 308
Gossard, E. E. 227, 268
Gozzini, A. 8, 21
Gray, R. E. 105, 171, 178, 226, 267
Grosskopf, J. F. 22, 171, 268
Groves, L. G. 6
Gunn, K. L. S. 274, 276, 291, 303, 308-309
H
Hasegawa, S. 47
Hathaway, S. D. 295, 299-302, 309
Hay, D. R. 36-37, 47, 171, 175, 224
Herbstreit, J. W. 31, 46, 148, 172, 374
Hewson, E. W. 268
Hill, R. M. 308
Hirai, M. 170, 225, 267
Hirao, K. 30, 45, 171, 373
Hogg, D. C. 238, 256, 258, 267
Holden, D. B. 182-183, 185, 227, 240,
266
Holmes, E. G. 46
Hopkins, H. G. 303, 309
Horn, J. D. 22, 171, 180, 226, 228, 267,
373
Hornberg, K. O. 374
Hughs, J. V. 21
Hull, R. A. 174, 225
Humphreys, W. J. 61, 88
Hurdis, E. C. 6, 21
I
Ikegami, F. 148, 172
Janes, H. B. 267, 374
Jehn, K. H. 19, 22, 171-172, 179-180,
193, 226-227, 239-240, 267
Johnson, W. E. 45, 170, 374
Jones, F. E. 45, 47
Jung, P. 47
K
Katz, I. 17, 22, 166, 171, 180, 193, 226,
239, 267, 312, 373
Kerker, M. 303, 309
Kerr, D. E. 141, 144, 171
Kirby, R. S. 148, 172
Kirkpatrick, A. W. 374
Kitchen, G. F. 268
Krinsky, A. 47
Kryder, S. J. 5, 22
Lane, J. A. 16, 22, 182, 227, 240, 266-267
Langleben, M. P. 303, 309
Laws, J. O. 292-293, 308
Lement, B. S. 46
List, R. J. 24, 45
Longley, A. 266
Longley, R. W. 268
Lukes, G. D. 166, 172, 180, 226
Lyons, H. 5, 22
M
Macready, P. B., Jr. 45
Magee, J. B. 8, 42
Martin, H. C. 47
Maryott, A. A. 5, 21, 273, 308
Mason, W. C. 22, 357, 374
Meadows, R. W. 268
Meaney, F. M. 266
Megaw, E. C. S. 227
Meyer, J. H. 255, 268
Middleton, W. E. K. 40, 47
Mie, G. 309
Millington, G. 13, 21
Minzner, R. A. 373
Misme, P. 30, 46, 147, 170-172, 175, 224-
225, 236, 238-240, 246, 265-267, 311,
373
Moler, W. F. 182-183, 185, 223, 227, 240,
266
Montgomery, R. B. 22, 267, 373
Mumford, W. W. 8, 22, 225
Murray, G. 47
N
Niwa, S. 170, 225, 267
Northover, F. H. 268
Norton, K. A. 62, 88, 105, 148, 171-172,
250, 266-267, 374
O
Onoe, M. 170, 225, 267
Ozanich, A. M., Jr. 171, 227, 373
Palmer, C. E. 201, 228
Parsons, D. A. 292-293, 308
Perlat, A. 224
Peterson, C. F. 374
Phillips, W. C. 6, 21
Pickard, G. W. 105, 170, 224, 230, 232,
235, 266
Pote, A. 308
Pound, R. V. 32, 36-37, 46
Price, W. L. 148, 172
R
Rainey, R. J. 374
Randall, D. L. 176-177, 224
Ratner, B. 168, 172, 280, 308, 373
Rice, P. L. 88, 92, 250, 266-267
Riggs, L. P. 22, 171, 180, 226, 228, 267
Ripley, W. S. 373
Roberts, C. S. 46
Rubenstein, P. J. 22, 267, 373
Ryde, D. 292, 294, 308
Ryde, J. W. 292, 294, 302-303, 308
Saito, S. 8, 21
Sargent, J. A. 31, 34, 46
Saxton, J. A. 1-2, 21, 182, 227, 238, 267-
268, 303, 309
Schelleng, J. C. 8, 21, 56, 88, 106, 171,
175, 225, 374
Schilkorski, W. 268
AUTHOR INDEX
435
Schulkin, M. 88, 150, 164, 170, 172, 179,
227, 373
Sheppard, P. A. 224
Silsbee, R. H. 47
Sion, E. 47
Smart, W. M. 49, 87, 172
Smith, E. K. 4, 6-8, 10-12, 17, 21
Smith, M. J. A. 47
Smith-Rose, R. L. 8, 22
Smyth, C. D. 21
Smyth, J. B. 6, 21, 227, 267
Snedecor, G. W. 171
Spilhaus, A. F. 40, 47
Spizzichino, A. 267
Sprung, A. 12, 21, 24, 45
Starkey, B. J. 268
Stetson, H. T. 105, 170, 224, 230, 232,
235, 266
Stickland, A. C. 8, 22, 61, 88
Stover, C. M. 45
Straiton, A. W. 30, 46, 274, 308
Stranathan, J. J. 6, 21
Strandberg, M. W. P. 308
Sugden, S. 6
Tao, K. 171, 373
Thayer, G. D. 22, 54, 87, 170, 225-226,
228, 267, 342, 363, 373-374, 377, 423
Thiesen, J. F. 30, 46
Thompson, M. C, Jr. 46, 374
Tinkham, M. 308
Tolbert, C. W. 274, 308
Trewartha, G. T. 172
Troitskii, V. S. 309, 312, 373
Trolese, L. G. 21, 227, 267
Turner, H. E. 47
Turner, W. R. 268
U
Unwin, R. S. 323
Van Vleck, J. H. 270, 273-275, 308
Vetter, M. J. 31, 34-36, 46, 374
Villars, F. 267
Voge, J. 238-239, 267
Yogler, L. E. 62, 88
W
Wadley, T. L. 8, 12-13, 21
Wagner, N. K. 38-39, 42, 47, 253, 268
Walkinshaw, W. 87, 171, 267
Waters, D. M. 46
Watkins, T. B. 47
Waynick, A. H. 8, 22
Weickmann, H. K. 291, 308
Weintraub, S. 4, 6-8, 10-12, 17, 21
Weisbrod, S. 80, 88
Weisskopf, V. F. 267
W^ells, P. I. 374
Wensien, H. 268
Wexler, A. 28-29, 39, 41, 45-47
Wilkerson, R. E. 374
Willett, H. C. 172
WiUiams, C. E. 46
Wilson, F. A. 22, 357, 374
Wolf, E. 21
Wong, M. S. 322, 373
Yerg, D. G. 45, 47, 174, 225
Zhevankin, S. A. 309, 312, 373
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